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All Textbook Solutions for Essentials of Statistics for the Behavioral Sciences

A researcher is interested in the texting habits of high school students in the United States. If the researcher measures the number of text messages that each individual sends each day and calculates the average number for the entire group of high school students, the average number would be an example of a ________________.A researcher is interested in how watching a reality television show featuring fashion models influences the eating behavior of 13-year-old girls. A group of 30 13-year-old girls is selected to participate in a research study. a. The group of 30 13-year-old girls is an example of a ________. b. In the same study, the amount of food eaten in one day is measured for each girl and the researcher computes the average score for the 30 13-year-old girls. The average score is an example of a ________.Statistical techniques are classified into two general categories. What are the two categories called, and what is the general purpose for the techniques in each category?Briefly define the concept of sampling error.A research study comparing alcohol use for college students in the United States and Canada reports that more Canadian students drink but American students drink more (Kuo, Adlaf, Lee, Gliksman, Demers, and Wechsler, 2002). Is this study an example of an experiment? Explain why or why not.What two elements are necessary for a research study to be an experiment?Stephens, Atkins, and Kingston (2009) conducted an experiment in which participants were able to tolerate more pain when they shouted their favorite swear words over and over than when they shouted neutral words. Identify the independent and dependent variables for this study.A tax form asks people to identify their annual income, number of dependents, and social security number. For each of these three variables, identify the scale of measurement that probably is used and identify whether the variable is continuous or discrete.An English professor uses letter grades (A, B, C, D, and F) to evaluate a set of student essays. What kind of scale is being used to measure the quality of the essays?The teacher in a communications class asks students to identify their favorite reality television show. The different television shows make up a scale of measurement.A researcher studies the factors that determine the number of children that couples decide to have. The variable, number of children, is a (discrete/ continuous) variable.a. When measuring height to the nearest inch, what are the real limits for a score of 68 inches? b. When measuring height to the nearest half inch, what are the real limits for a score of 68 inches?Calculate each value requested for the following scores: 4, 3, 7, 1. a. X b. X2 c. (X)2 d. X1 e. (X1) f. (X1)2 2LC Use summation notation to express each of the following. a. Subtract 2 points from each score and then add the resulting values. b. Subtract 2 points from each score, square the resulting values, and then add the squared numbers. c. Add the scores and then square the total.A researcher is investigating the effectiveness of a treatment for adolescent boys who are taking medication for depression. A group of 30 boys is selected and half receive the new treatment in addition to their medication and the other half continue to take their medication without any treatment. For this study, a. Identify the population. b. Identify the sample.Define the terms population, sample, parameter, and statistic.Statistical methods arc classified into two major categories: descriptive and inferential. Describe the general purpose for the statistical methods in each category.Define the concept of sampling error and explain why this phenomenon creates a problem to he addressed by inferential statistics.Describe the data for a correlation research study and explain how these data are different from the data obtained in experimental and nonexperimental studies, which also evaluate relationships between two variables.What is the goal for an experimental research study? Identify the two elements that are necessary for an experiment to achieve its goal.Knight and Haslam (2010) found that office workers who had sonic input into the design of their office space were more productive and had higher wellbeing compared to workers for whom the office design was completely controlled by an office manager. For this study, identify the independent variable and the dependent variable.Judge and Cable (2010) found that thin women had higher incomes than heavier women. Is this an example of an experimental or a nonexperimental study?Two researchers are both interested in determining whether large doses of vitamin C can help prevent the common cold. Each obtains a sample of n = 20 college students. a. The first researcher interviews each student to determine whether they routinely take a vitamin C supplement. The researcher then records the number of colds each individual gets during the winter. Is this an experimental or a nonexperimental study? Explain your answer. b. The second researcher separates the students into two roughly equivalent groups. The students in one group are given a daily multivitamin containing a large amount of vitamin C, and the other group gets a multivitamin with no vitamin C. The researcher then records the number of colds each individual gets during the winter. Is this an experimental or a nonexperimental study? Explain your answer.Weinstein, McDermott, and Roediger (2010) conducted an experiment to evaluate the effectiveness of different study strategies. One part of the study asked students to prepare for a test by reading a passage. In one condition, students generated and answered questions after reading the passage. In a second condition, students simply read the passage a second time. All students were then given a test on the passage material and the researchers recorded the number of correct answers. a. Identify the dependent variable for this study. b. Is the dependent variable discrete or continuous ? c. What scale of measurement (nominal, ordinal, interval, or ratio) is used to measure the dependent variable?A research study reports that alcohol consumption is significantly higher for students at a state university than for students at a religious college (Wells, 2010). Is this study an example of an experiment? Explain why or why not.Oxytocin is a naturally occurring brain chemical that is nicknamed the love hormone because it seems to play a role in the formation of social relationship? such as mating pairs and parent-child bonding. A recent study demonstrated that oxytocin appears to increase peoples tendency to trust others (Kosfeld, Heinrichs, Zak, Fischbacher, and Fehr, 2005). Using an investment game, the study demonstrated that people who inhaled oxytocin were more likely to give their money to a trustee compared to people who inhaled an inactive placebo. For this experimental study, identify the independent variable and the dependent variable.For each of the following, determine whether the variable being measured is discrete or continuous and explain your answer. a. Social networking (number of daily minutes on Facebook) b. Family size (number of siblings) c. Preference between digital or analog watch d. Number of correct answers on a statistics quiz Four scales of measurement were introduced in this chapter: nominal, ordinal, interval, and ratio. a. What additional information is obtained from measurements on an ordinal scale compared to measurements on a nominal scale? b. What additional information is obtained from measurements on an interval scale compared to measurements on an ordinal scale? c. What additional information is obtained from measurements on a ratio scale compared to measurements on an interval scale?In an experiment examining the effects Tai Chi on arthritis pain. Callahan (2009) selected a large sample of individuals with doctor-diagnosed arthritis. Half of the participants immediately began a Tai Chi course and the other half (the control group) waited 8 weeks before beginning the program. At the end of 8 weeks, the individuals who had experienced Tai Chi had less arthritis pain that those who had not participated in the course. a. Identify the independent variable for this study. b. What scale of measurement is used for the independent variable? c. Identify the dependent variable for this study. d. What scale of measurement is used for the dependent variable?Explain why shyness is a hypothetical construct instead of a concrete variable. Describe how shyness might be measured and defined using an operational definition.Ford and Torok (2008) found that motivational signs were effective in increasing physical activity on a college campus. Signs such as Step up to a healthier lifestyle and An average person bums 10 calories a minute walking up the stairs were posted by the elevators and stairs in a college building. Students and faculty increased their use of the stairs during times that the signs were posted compared to times when there were no signs. a. Identify the independent and dependent variables for this study. b. What scale of measurement is used for the independent variable?For the following scores, find the value of each expression: a. X b. X2 c. X + 1 d. (X + 1) X 3 5 0 2 For the following set of scores, find the value of each expression: a. X2 b. (X)2 c. (X 1) d. (X 1)2 X 3 2 5 1 3 For the following set of scores, find the value of each expression: a. X b. X2 c. (X +3) X 6 -2 0 -3 -1 Two scores, X and Y, are recorded for each of n = 4 subjects. For these scores, find the value of each expression. a.X b.Y c.XY Subject X Y A 3 4 B 0 7 C -1 5 D 2 2 Use summation notation to express each of the following calculations: a. Add 1 point to each score, and then add the resulting values. b. Add 1 point to each score and square the result. Then add the squared values. c. Add the scores and square the sum. Then subtract 3 points from the squared value.23P Construct a frequency distribution table for the following set of scores. Scores: 3, 2,3, 2, 4, 1,3, 3,5 Find each of the following values for the sample in the following frequency distribution table. a.n b.X c.X2 X f 5 1 4 2 3 2 2 4 1 I 1LCA 2LCA 3LCA Sketch a histogram and a polygon showing the distribution of scores presented in the following table: X f 7 1 6 1 5 3 4 6 3 4 2 1 Describe the shape of the distribution in Exercise 1. Exercise 1: histogram 3LC If the results from a research study are presented in a frequency distribution histogram, would it also be appropriate to show the same results in a polygon? Explain your answer.5LC Place the following set of n = 20 scores in a frequency distribution table. 6 2 2 1 3 7 4 7 1 2 5 3 1 6 2 6 3 3 7 2 Construct a frequency distribution table for the following set of scores. Include columns for proportion and percentage in your tables. Scores: 5 7 8 4 7 9 6 6 5 3 9 6 4 7 7 8 6 7 8 5 Find each value requested for the distribution of scores in the following table. a. n b. X c. X2 X f 5 2 4 3 3 5 2 1 1 1 Find each value requested for the distribution of scores in the following table. a. n b. X c. X2 X f 5 1 4 2 3 3 2 5 1 3 For the following scores, the smallest value is X = 17 and the largest value is X = 53. Place the scores in a grouped frequency distribution table a. using an interval width of 5 points. b. using an interval width of 10 points. 44 19 23 17 25 47 32 26 25 30 18 24 49 51 24 19 43 27 34 18 52 18 36 25 The following scores are the ages for a random sample of n = 30 drivers who were issued speeding tickets in New York during 2008. Determine the best interval width and place the scores in a grouped frequency distribution table. Prom looking at your table, does it appear that tickets are issued equally across age groups? 17 30 45 20 39 53 28 19 24 21 34 38 22 29 64 22 44 36 16 56 20 23 58 32 25 28 22 51 26 43 For each of the following samples, determine the interval width that is most appropriate for a grouped frequency distribution and identify the approximate number of intervals needed to cover the range of scores. a. Sample scores range from X = 8 to X = 41. b. Sample scores range from X = 16 to X = 33. c. Sample scores range from X = 26 to X = 98.What information is available about the scores in a regular frequency distribution table that you cannot obtain for the scores, in a grouped table?Describe the difference in appearance between a bar graph and a histogram and describe the circumstances in which each type of graph is used.For the following set of scores: 8 5 9 6 8 7 4 10 6 7 9 7 9 9 5 8 8 6 7 10 a. Construct a frequency distribution table to organize the scores. b. Draw a frequency distribution histogram for these data.Sketch a histogram and a polygon showing the distribution of scores presented in the following table: X f 7 1 6 1 5 3 4 6 3 4 2 1 Sketch a histogram showing the distribution of scores shown in the following table: X f 45-49 4 40-44 6 35-39 10 30-34 5 25-29 3 20-24 2 A survey given to a sample of college students contained questions about the following variables. For each variable, identify the kind of graph that should be used to display the distribution of scores (histogram. polygon, or bar graph). a. number of brothers and sisters b. birth-order position among siblings (oldest = 1st) c. gender (male/female) d. favorite television show during the previous year Each year the college gives away T shirts to new students during freshman orientation. The students are allowed to pick the shirt sizes that they want. To determine how many of each size shirt they should order, college officials look at the distribution from last year. The following table shows the distribution of shirt sizes selected last year. Size f S 27 M 48 L 136 XL 120 XXI. 39 a. What kind of graph would be appropriate for showing this distribution? b. Sketch the frequency distribution graph.Gaucher, Friesen, and Kay (2011) found that words they identified as masculine themed (such tut competitive, independent, analyze, strong) are commonly used in job recruitment materials, especially for job advertisements in male-dominated areas. In a similar study, a researcher counted the number of masculine-themed words in job advertisement for job areas, and obtained the following data. Area Number of Masculine Words Plumber 14 Electrician 12 Security guard 17 Bookkeeper 9 Nurse 6 Early-childhood educator 7 Determine what kind of graph would be appropriate for showing this distribution and sketch the frequency distribution graph.Find each or the followin.tt values for the distribution shown in the following polygon. a. n b. X c. X2 17P Place the following scores in a frequency distribution table. Based on the frequencies, what is the shape of the distribution? 13 14 12 15 15 14 15 11 13 14 11 13 15 12 14 14 10 14 13 15 For the following set of scores: 8 6 7 5 4 10 8 9 5 7 2 9 9 10 7 8 8 7 4 6 3 8 9 6 a. Construct a frequency distribution table. b. Sketch a histogram showing the distribution. c. Describe the distribution using the following characteristics: (1) What is the shape of the distribution? (2) What score best identifies the center (average) for the distribution? (3) Are the scores clustered together, or are they spread out across the scale?Fowler and Christakis (2008) report that personal happiness tends to be associated with having a social network including many other happy friends. To test this claim, a researcher obtains a sample of n = 16 adults who claim to be happy people and a similar sample of n = 16 adults who describe themselves as neutral or unhappy. Each individual is then asked to identify the number of their close friends whom they consider to be happy people. The scores are as follows: Happy: 8 7 4 10 6 6 8 9 8 8 7 5 6 9 8 9 Unhappy: 5 8 4 6 6 7 9 6 2 8 5 6 4 7 5 6 Sketch a polygon showing the frequency distribution for the happy people. In the same graph, sketch a polygon for the unhappy people. (Use two different colors, or use a solid line for one polygon and a dashed line for the other.) Does one group seem to have more happy friends?Recent research suggests that the amount of lime that parents spend talking about numbers can have a big impact on the mathematical development of their children (Levine, Suriyakham, Rowe, Huttenlocher, Gunderson. 2010). In the study, the researchers visited the childrens homes between the ages of 14 and 30 months and recorded the amount of number talk they heard from the childrens parents. The researchers then tested the childrens knowledge of the meaning of numbers at 46 months. The following data are similar to the results obtained in the study. Childrens Knowledge of-Numbers Scores for Two Groups of Parents Low Number Talk Parents High Number Talk Parents 2, 1, 2, 3, 4 3, 4, 5, 4, 5 3, 3, 2, 2, 1 4, 2, 3, 5, 4 5, 3, 4, 1, 2 5, 3, 4, 5, 4 Sketch a polygon showing the frequency distribution for children with low number-talk parents. In the same graph, sketch a polygon showing the scores for the children with high number-talk parents. (Use two different colors or use a solid line for one polygon and a dashed line for the other.) Does it appear that there is a difference between the two groups?Find the mean for the following sample of n = 5 scores: 1, 8, 7, 5, 9.A sample of n = 6 scores has a mean of M = 8. What is the value of X for this sample?3LC A sample of n = 6 scores has a mean of M = 40. One new score is added to the sample and the new mean is found to be M = 35. What can you conclude about the value of the new score? a. It must be greater 40. b. It must be less than 40.5LC 1LCA 2LCA A population has a mean of = 40. a. If 5 points were added to every score, what would be the value for the new mean? b. If every score were multiplied by 3, what would be the value for the new mean?A sample of n = 4 scores has a mean of 9. If one person with a score of X = 3 is removed from the sample, what is the value for the new sample mean?1LC If you have a score of 52 on an 80-point exam, then you definitely scored above the median. (True or false?)The following is a distribution of measurements for a continuous variable. Find the precise median that divides the distribution exactly in half. Scores: 1, 2, 2, 3, 4, 4, 4, 4, 4, 5 During the month of October, an instructor recorded the number of absences for each student in a class of n = 20 and obtained the following distribution. Number of Absences f 5 1 4 2 3 7 2 5 1 3 0 2 a. Using the mean, what is the average number of absences for the class? b. Using the median, what is the average number of absences tor the class? c. Using the mode, what is the average number of absences for the class?Which measure of central tendency is most affected if one extremely large score is added to a distribution? (mean, median, mode)What is it usually considered inappropriate to compute a mean for scores measured on an ordinal scale?3LC A distribution with a mean of 70 and a median of 75 is probably positively skewed. (True or false?)Why is it necessary to have more than one method for measuring central tendency?Find the mean, median, and mode for the following sample of scores: 5 4 5 2 7 1 3 5 Find the mean, median, and mode for the following sample of scores: 3 6 7 3 9 8 3 7 5 Find the mean, median, and mode for the scores in the following frequency distribution table: X f 6 1 5 2 4 2 3 2 2 2 1 5 Find the mean, median, and mode for the scores in the following frequency distribution table: X f 8 1 7 1 6 2 5 5 4 2 3 2 For the following sample: a. Assume that the scores are measurements of a continuous variable and find the median by locating the precise midpoint of the distribution. b. Assume that the scores are measurements of a discrete variable and find the median. Scores: 123334 A population of N = 15 scores has X = 120. What is the population mean?A sample of n = 8 scores has a mean of M = 12. What is the value of X for this sample?A population with a mean of = 8 has X = 40. How many scores are in the population?A sample of n = 7 scores has a mean of M = 9. If one new person with a score of X = 1 is added to the sample, what is the value for the new mean?A sample of n = 6 scores has a mean of M = 13. If one person with a score of X = 3 is removed from the sample, what is the value for the new mean?A sample of n = 15 scores has a mean of M = 6. One person with a score of X = 22 is added to the sample. What is the value for the new sample mean?A sample of n = 10 scores has a mean of M = 9. One person with a score of X = 0 is removed from the sample. What is the value for the new sample mean?A population of N = 15 scores has a mean of = 8. One score in the population is changed from X = 20 to X = 5. What is the value for the new population mean?A sample of n = 7 scores has a mean of M = 16. One score in the sample is changed from X = 6 to X = 20. What is the value for the new sample mean?A sample of n = 7 scores has a mean of M = 5. After one new score is added to the sample, the new mean is found to be M = 6. What is the value of the new score? (Hint: Compare the values for X before and after the score was added.)A population of N = 8 scores has a mean of = 16. After one score is removed from the population, the new mean is found to be = 15. What is the value of the score that was removed? (Hint: Compare the values for X before and after the score was removed.)A sample of n = 9 scores has a mean of M = 13. After one score is added to the sample, the mean is found to be M = 12. What is the value of the score that was added?A sample of n = 9 scores has a mean of M = 20. One of the scores is changed and the new mean is found to be M = 22. If the changed score was originally X = 7, what is its new value?One sample of n = 12 scores has a mean of M = 7 and a second sample of n = 8 scores has a mean of M = 12. If the two samples are combined, what is the mean for the combined sample?One sample has a mean of M = 8 and a second sample has a mean of M = 16. The two samples are combined into a single set of scores. a. What is the mean for the combined set if both of the original samples have n = 4 scores? b. What is the mean for the combined set if the first sample has n = 3 and the second sample has n = 5? c. What is the mean for the combined set if the first sample has n = 5 and the second sample has n = 3?One sample has a mean of M = 5 and a second sample has a mean of M = 10. The two samples are combined into a single set of scores. a. What is the mean for the combined set if both of the original samples have n = 5 scores? b. What is the mean for the combined set if the first sample has n = 4 scores and the second sample has n = 6? c. What is the mean for the combined set if the first sample has n = 6 scores and the second sample has n = 4?Explain why the mean is often not a good measure of central tendency for a skewed distribution.A researcher conducts a study comparing two different treatments with a sample of n = 16 participants in each treatment. The study produced the following data: Treatment 1: 6 7 11 4 19 17 2 5 9 13 6 23 11 4 6 1 Treatment 2: 10 9 6 61 11 8 6 3 2 11 1 12 7 109 a. Calculate the mean for each treatment. Based on the two means, which treatment produces the higher scores? b. Calculate the median for each treatment. Based on the two medians, which treatment produces the higher scores? c. Calculate the mode for each treatment. Based on the two modes, which treatment produces the higher scores?Schmidt (1994) conducted a series of experiments examining the effects of humor on memory. In one study, participants were shown a list of sentences, of which half were humorous and half were non- humorous. A humorous example is, If at first you don't succeed, you are probably not related to the boss." Other participants would see a non-humorous version of this sentence, such as People who are related to the boss often succeed the very first time. Schmidt then measured the number of each type of sentence recalled by each participant. The following scores are similar to the results obtained in the study. Number of Sentences Recalled Humorous Sentences Nonhumorous Sentences 4 5 2 4 5 2 4 2 6 7 6 6 2 3 1 6 2 5 4 3 3 2 3 3 1 3 5 5 4 1 5 3 Calculate the mean number of sentences recalled for each of the two conditions. Do the data suggest that humor helps memory?Stephens, Atkins, and Kingston (2009) conducted a research study demonstrating that swearing can help reduce pain. In the study, each participant was asked to plunge a hand into icy water and keep it there as long as the pain would allow. In one condition, the participants repeatedly yelled their favorite curse words while their hands were in the water. In the other condition the participants repeated a neutral word. Data similar to the results obtained in the study are shown in the following table. Calculate the mean number of seconds that the participants could tolerate the pain for each of the two treatment conditions. Does it appear that swearing helped with pain tolerance? Amount of Time (in seconds) Participant Swear Words Neutral Words 1 94 59 2 70 61 3 52 47 4 83 60 5 46 35 6 117 92 7 69 53 8 39 30 9 51 56 10 73 61 Earlier in this chapter (p. 67), we mentioned a research study demonstrating that alcohol consumption increases attractiveness ratings for members of the opposite sex (Jones. Jones, Thomas. Piper, 2003). In the actual study, college-age participants were recruited from bars and restaurants near campus and asked to participate in a market research" study. During the introductory conversation, they were asked to report their alcohol consumption for the day and were told that moderate consumption would not prevent them from taking part in the study. Participants were then shown a series of photographs of male and female faces and asked to rate the attractiveness of each face on a 1-7 stale. The following data duplicate the general pattern of results obtained in the study. The two sets of scores are attractiveness ratings tor one female obtained from two groups of males: those who had no alcohol and those with moderate alcohol consumption. Calculate the mean for each group. Docs it appear from these data that alcohol has an effect on judgments of attractiveness? Croup J No Alcohol Croup 2 Moderate Alcohol 3 4 5 1 2 53524 42 344 6 5 6 5 4 5 6 3 4 3 7 5 65 6 Brief explain what is measured by the standard deviation and what is measured by the variance.The deviation scores are calculated for each individual in a population of N = 4. The first three individuals have deviations of +2, +4, and 1. What is the deviation for the fourth individual?What is the standard deviation for the following set of N = 5 scores: 10, 10, 10, 10, and 10? (Note: You should be able to answer this question directly from the definition of standard deviation, without doing any calculations.)Calculate the variance for the following population of N = 5 scores: 4, 0, 7, 1, 3.Find the sum of the squared deviations, SS, for each of the following populations. Note that the definitional formula works well for one population but the computational formula is better for the other. Population 1: 3 1 5 1 Population 2: 6 4 2 0 9 3 a. Sketch a histogram showing the frequency distribution for the following population of N = 6 scores: 12, 0, 1. 7, 4, 6. Locate the mean in your sketch, and estimate the value of the standard deviation. b. Calculate SS, variance, and the standard deviation for these scores. How well does your estimate compare with the actual standard deviation?a. Sketch a histogram showing the frequency distribution for the following sample of n = 5 scores: 3, 1, 9, 4, 3. Locale the mean in your sketch, and estimate the value of the sample standard deviation. b. Calculate SS, variance, and standard deviation for this sample. How well does your estimate from part a compare with the real standard deviation?For the following set of scores: 1, 5, 7, 3, 4 a. Assume that this is a population of N = 5 scores and compute SS and variance for the population. b. Assume that this is a sample of n = 5 scores and compute SS and variance for the sample.3LC Explain the difference between a biased and an unbiased statistic.In a population with a mean of = 50 and a standard deviation of = 10, would a score of X = 58 be considered an extreme value (far out in the tail of the distribution)? What if the standard deviation were = 3?A population has a mean of . = 70 and a standard deviation of = 5. a. If 10 points were added to every score in the population, what would be the new values for the population mean and standard deviation? b. If every score in the population were multiplied by 2, what would be the new values for the population mean and standard deviation?In words, explain what is measured by each of the following: a. SS b. Variance c. Standard deviation Can SS ever have a value less than zero? Explain your answer.Is it possible to obtain a negative value for the variance or the standard deviation?What does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample.Explain why the formula for sample variance is different from the formula for population variance.A population has a mean of = 80 and a standard deviation of = 20. a. Would a score of X = 70 be considered an extreme value (out in the tail) in this sample? b. If the standard deviation were = 5, would a score of X = 70 be considered an extreme value?On an exam with a mean of M = 78, you obtain a score of X = 84. a. Would you prefer a standard deviation of s = 2 or s = 10? (Hint: Sketch each distribution and find the location of your score.) b. If your score were X = 72, would you prefer s = 2 or s = 10? Explain your answer.Calculate the mean and SS (sum of squared deviations) tor each of the following samples. Based on the value for the mean, you should be able to decide which SS formula is better to use. Sample A: 1 4 8 5 Sample B: 3 0 9 4 For the following population of N = 6 scores: 3 1 4 3 3 4 a. Sketch a histogram showing the population distribution. b. Locate the value of the population mean in your sketch, and make an estimate of the standard deviation (as done in Example 4.2). c. Compute SS, variance, and standard deviation for the population. (How well does your estimate compare with the actual value of ?)For the following sample of n = 7 scores: 8 6 5 2 6 3 5 a. Sketch a histogram showing the sample distribution. b. Locate the value of the sample mean in your sketch, and make an estimate of the standard deviation (as done in Example 4.5). c. Compute SS, variance, and standard deviation for the sample. (How well does your estimate compare with the actual value of .s?)For the following population of N = 6 scores: 11 0 2 9 9 5 a. Calculate the range and the standard deviation. (Use either definition for the range.) b. Add 2 Point to each score and compute the range and standard deviation again. Describe how adding a constant to each score influences measures of variability.The range is completely determined by the two extreme scores in a distribution. The standard deviation, on the other hand, uses every score. a. Compute the range (choose either definition) and the standard deviation for the following sample of n = 5 scores. Note that there are three scores clustered around the mean in the center of the distribution, and two extreme values. Scores: 067814 a. Now we break up the cluster in the center of the distribution by moving two of the central scores out to the extremes. Once again compute the range and the standard deviation. New scores:0071414 a. According to the range, how do the two distributions compare in variability? How do they compare according to the standard deviation?A population has a mean of = 30 and a standard deviation of = 5. a. If 5 points were added to every score in the population, what would be the new values for the mean and standard deviation? b. If every score in the population were multiplied by 3, what would he the new values for the mean and standard deviation?a. After 3 points have heen aikicd to every score in a sampe, the mean is found to be M = 83 and the standard deviation is s = 8. What were the values for the mean and standard deviation for the Original sample? b. After every score in a sample has been multiplied by 4, the mean is found to be M = 48 and the standard deviation is s = 12. What were the values for the mean and standard deviation for the originai sample?For the following sample of n = 4 scores: 82, 88, 82, and 86: a. Simplify the arithmetic by first subtracting 80 points from each score to obtain a new sample of 2, 8, 2, and 6. Then, compute the mean and standared deviation for the new sample. b. Using the values you obtained in part a, what are the values for the mean and standard deviation for the original sample?For the following sample of n = 8 scores: 0, 1, 1/2, 0, 3, 1/2, 0, and 1: a. Simplify the arithmetic by first multiplying each score by 2 to obtain a new sample of 0, 2, 1, 0, 6, 1, 0, and 2. Then, compute the mean and standard deviation for the new sample. b. Using the values you obtained in part a, what are the values for the mean and standard deviation for the original sample?For the data in the following sample: 81515 a. Find the mean and the standard deviation. b. Now change the score of X = 8 to X = 18, and find the new mean and standard deviation. c. Describe how one extreme score influences the mean and standard deviation.Calculate the mean and SS (sum of squared deviations) for each of the following samples. Based on the value for the mean, you should be able to decade which SS formula is better to use. Sample A: 1 4 8 5 Sample B: 3 0 9 4 Calculate SS, variance, and standard deviation for the following population of N = 8 scores: 0, 0, 5, 0, 3, 0, 0, 4. (Note: The computational formula for SS works well with these scores.)Calculate SS, variance, and standard deviation for the following population of N = 6 scores: 1, 6, 10, 9, 4, 6. (Note: The definitional formula for SS works well with these scores.)Calculate SS, variance, and standard deviation for the following sample of n = 5 scores: 10, 4, 8, 5, 8. (Note: The definitional formula for SS works well with these scores.)In an extensive study involving thousands of British children. Arden and Plomin (2006) found significantly higher variance in the intelligence scores for males than for females. Following are hypothetical data, similar to the results obtained in the study. Note that the scores are not regular IQ scores but have been standardized so that the entire sample has a mean of M = 10 and a standard deviation of s = 2. a. Calculate the mean and the standard deviation for the sample of n = 8 females and for the sample of n = 8 males. b. Based on the means and the standard deviations, describe the differences in intelligence scores for males and females. Female Male 9 8 11 10 10 11 13 12 8 6 9 10 11 14 9 9 Within a population, the difference that exist from one person to another are often called diversity. Researchers comparing cognitive skills for younger adults and older adults, typically find greater difference (greater diversity) in the older population (Morse. 1993). Following are typical data showing problem-Solving scores for two groups of participants. Older Adults (average age 72) Younger Adults (average age 31) 9 4 7 3 8 7 9 6 7 8 6 2 8 5 4 6 7 6 6 8 7 5 2 6 6 9 7 8 6 9 a. Compute the mean, the variance, and the standard deviation for each group. b. Is one group or scores noticeably more variable (more diverse) than the other?In the previous problem we noted that the differences in cognitive skills tend to be bigger among older people than among younger people. These differences are often called diversity. Similarly, the differences in performance from trial to trial for the same person are often called consistency. Research in this area suggests that consistency of performance seems to decline as people age. A study by Wegesin and Stem (2004) found lower consistency (more variability) in the memory performance scores for older women than for younger women. The following data represent memory scores obtained for two women, one older and one younger, over a series of memory trials. a. Calculate the variance of the scores for each woman. b. Are the scores for the younger woman more consistent (less variable)? Younger Older 8 7 6 5 6 8 7 5 8 7 7 6 8 8 8 5 a. What is the general goal for descriptive statistics? b. How is the goal served by putting scores in a frequency distribution? c. How is the goal served by computing a measure of central tendency? d. How is the goal served by computing a measure of variability?Identify the z-score value corresponding to each of the following locations in a distribution. a. Below the mean by 2 standard deviations. b. Above the mean by standard deviation. c. Below the mean by 1.50 standard deviations.Describe the location in the distribution for each of the following z-scores. (For example, z = +1.00 is located above the mean by 1 standard deviation.) a. z = 1.50 b. z = 0.25 c. z = 2.50 d. z = 0.50 For a population with = 30 and = 8, find the z-score for each of the following scores: a. X = 32 b. X = 26c. X = 42 4LC 1LCA 2LCA 3LCA In a distribution with = 12, a score of X = 56 corresponds to z = 0.25. What is the mean for this distribution?A normal-shaped distribution with = 40 and = 8 is transformed into z-scores. Describe the shape, the mean, and the standard deviation for the resulting distribution of z-scores.What is advantage of having a mean of = 0 for a distribution of z -scores?A distribution of English exam scores has = 70 and = 4. A distribution of history exam scores has = 60 and = 20. For which exam would a score of X = 78 have a higher standing? Explain your answer.4LC A population of scores has = 73 and = 8. If the distribution is standardized to create a new distribution with = 100 and = 20, what are the new values for each of the following scores from the original distribution? a. X = 65b. X = 71c. X = 81d. X = 83 2LC For a sample with a mean of M = 40 and a standard deviation of s = 12, find the z-score corresponding to each of the following X values. X = 43 X = 58 X = 49 X = 34 X = 28 X = 16 For a sample with a mean of M = 80 and a standard deviation of s = 20, find the X value corresponding to each of the following z-scores. z = 1.00 z = 0.50 z = 0.20 z= 1 50 z = 0.80 z = 1.40 3LC For a sample with a standard deviation of s = 12, a score of X = 83 corresponds to z = 0.50. What is the mean for the sample?A sample has a mean of M = 30 and a standard deviation of s = 8. a. Would a score of X = 36 be considered a central score or an extreme soon the sample? b. If the standard deviation were s = 2, would X = 36 be central or extreme What information is provided by the sign (+/) of a z-score? What information is provided by the numerical value of the z-score?A distribution has a standard deviation of = 10. Find the c-score for each of the following locations in the distribution. a. Above the mean by 5 points. b.Above the mean by 2 points. c. Below the mean by 20 points. d. Below the mean by 15 points.For a distribution with a standard deviation of = 20, describe the location of each of the following z-scores in terms of its position relative to the mean. For example, z = +1.00 is a location that is 20 points above the mean. a. z = +2.00 b. z=+0.50 c.z=1.00 d.z =0.25 For a population with = 80 and = 10, a.Find the z-score for each of the following X values. (Note: You should be able to find these values using the definition of a z-score. You should not need to use a formula or do any serious calculations.) X = 75 X = 100 X = 60 X = 95 X = 50 X = 85 For a population with = 40 and a = 11, find the z-score for each of the following X values. (Note: You probably will need to use a formula and a calculator to find these values.) X = 45 X = 52 X = 41 X = 30 X = 25 X = 38 For a population with a mean of = 100 and a standard deviation of = 20, a. Find the z-score for each of the following X values. X = 108 X = 115 X = 130 X = 90 X = 88 X = 95 b. Find the score (X value) that corresponds to each of the following c-scores. z= 0.40 z= 0.50 z= 1.80 z = 0.75 z = 1-50 z = -1.25 A population has a mean of = 60 and a deviation of = 12. a. For this population, find the z-score for each of the following X values. X = 69 X = 84 X = 63 X = 54 X= 48 X = 45 b. For the same population, find the score (X value) that corresponds, to each of the following z-scores. z = 0.50 z = 1.50 z = 2.50 z = 0.25 z = 0.50 z = 1.25 A sample has a mean of M = 30 and a standard deviation of s = 8. Find the s-score for each of the following X values from this sample. X = 32 X = 34 X = 36 X = 28 X = 20 X = 18 A sample has a mean of M = 25 and a standard deviation of s = 5. For this sample, find the X value corresponding to each of the following z-scores. z = 0.40 z = 1.20 z = 2.00 z = 0.80 z = 0.60 z = 1.40 Find the z-score corresponding to a score of X = 45 for each of the following distributions. a. = 40 and = 20 b. = 40 and = 10 c. = 40 and = 5 d. = 40 and = 2 Find the X value corresponding to z = 0.25 for each of the following distributions. a. = 40 and = 4 b. = 40 and = 8 c. = 40 and = 16 d. = 40 and =32 A score that is 6 points below the mean corresponds to a z-score of z = 2.00. What is the population standard deviation?A score that is 9 points above the mean corresponds to z-score of z = 1 50. What is the population standard deviation?For a population with a standard deviation of = 12, a score of X = 44 corresponds to z = 0.50. What is the population mean?For a sample with a standard deviation of s = 8, a score of X = 65 corresponds to z = 1.50. What is the sample mean?For a sample with a mean of M = 51, a score of X = 59 corresponds to z = 2.00. What is the sample standard deviation?For a population with a mean of = 70, a score X = 64 corresponds to z = 1.50. What is the population standard deviation?In a population distribution, a score of X = 28 corresponds to z = 1.00 and a score of X = 34 corresponds to z = 0.50. Find the mean and standard deviation for the population. (Hint: Sketch the distribution and locate the two scores on your sketch.)In a sample distribution, X = 56 corresponds to z = 1.00, and X = 47 corresponds to z= 0.50. Find the mean and standard deviation for the sample.For each of the following populations, would a score of X = 50 be considered a central score (near the middle of the distribution) or an extreme score (far out in the tail of the distribution)? a. = 45 and = 10 b. = 45 and = 2 c. = 90 and = 20 d. = 60 and = 20 A distribution of exam scores has a mean of = 78. a. If your score is X = 70, which standard deviation would give you a better grade: = 4 or = 8? b. If your score is X = 80, which standard deviation would give you a better grade: = 4 or = 8?For each of the following, identify the exam score that should lead to the better grade. In each case, explain your answer. a. A score of X = 74 on an exam with M = 82 and = 8; or a score of X = 40 on an exam with = 50 and = 20. b. A score of X = 51 on an exam with = 45 and = 2: or a score of X = 90 on an exam with = 70 and = 20. c. A score of X - 62 on an exam with = 50 and = 8; or a score of X = 23 on an exam with = 20 and = 2.A distribution with a mean of = 38 and a standard deviation of = 5 is transformed into a standardized distribution with = 50 and = 10. Find the new, standardized score for each of the following values from the original population. a.X = 39 b.X = 43 c.X = 35 d.X = 28 A distribution with a mean of = 76 and a standard deviation of = 12 is transformed into a standardized distribution with = 100 and = 20. Find the new, standardized score for each of the following values from the original population. a.X = 61 b. X = 70 c.X = 85 d. X = 94 A population consists of the following N = 5 scores: 0, 6, 4, 3. and 12. a. Compute and for the population. b. Find the z-score for each score in the population. c. Transform the original population into a new population of N = 5 scores with a mean of = 100 and a standard deviation of = 20.A sample consists of the following n = 7 scores: 5, 0, 4.5, 1.2, and 4. a. Compute the mean and standard deviation for the sample. b. Find the z-score for each score in the sample. c. Transform the original sample into a new sample with a mean of M = 50 and s=10.A survey of the students in a psychology class revealed that there were 19 females and 8 males. Of the 19 females, only 4 had no brothers or sisters, and 3 of the males were also the only child in the household. If a student is randomly selected from this class, a. What is the probability of obtaining a male? b. What is the probability of selecting a student who has at least one brother or sister? c. What is the probability of selecting a female who has no siblings?A jar contains 10 red marbles and 30 blue marbles. a. If you randomly select 1 marble from the jar, what is the probability of obtaining a red marble? b. If you take a random sample of n = 3 marbles from the jar and the first two marbles are both blue, what is the probability that the third marble will be red?Suppose that you are going to select a random sample of n = 1 score from the distribution in Figure 6.2. Find the following probabilities: a. p(X 2) b. p(X 5) c. p(X 3)Find the proportion of a normal distribution that corresponds to each of the following sections: a. z 0.25 b. z 0.80 c.z 1.50 d.z 0.75 For a normal distribution, find the z-score location that divides the distribution as follows: a.Separate the top 20% from the rest. b. Separate the top 60% from the rest. c. Separate the middle 70% from the rest.3LC For a normal distribution with a mean of = 60 and a standard deviation of = 12, find each probability value requested. a. p(X 66) b. p(X 75) cp(X 57) d. p(48 X 72)Scores on the Mathematics section of the SAT Reasoning Test form a normal distribution with a mean of = 500 and a standard deviation of = 100. a. If the state college only accepts students who score in the lop 60% on this test, what is the minimum score needed for admission? b. What is the minimum score necessary to be in the lop 10% of the distribution? c. What scores form the boundaries for the middle 50% of the distribution?What is the probability of selecting a score greater than 45 from a positively skewed distribution with = 40 and = 10? (Be careful.)A local hardware store has a Savings Wheel at the checkout. Customers get to spin the wheel and. when the wheel stops, a pointer indicates how much they will save. The wheel can stop in any one of 50 sections. Of the sections, 10 produce 0% off, 20 sections are for 10%off, 10 sections for 20%, 5 for 30%, 3 for 40%, 1 for 50%, and 1 for 100% off. Assuming that all 50 sections are equally likely, a. What is the probability that a customer's purchase will be free (100% off)? b. What is the probability that a customer will get no savings from the wheel (0% off)? c. What is the probability that a customer will get at least 20% off?A psychology class consists of 14 males and 36 females. If the professor selects names from the class list using random sampling, a. What is the probability that the first student selected will be a female? b. If a random sample of n = 3 students is selected and the first two are both females, what is the probability that the third student selected will be a male?What are the two requirements for a random sample?Draw a vertical line through a normal distribution for each of the following z-score locations. Determine whether the tail is on the right or left side of the line and find the proportion in the tail. a. z = 1.00 b. z = 0.50 c. z= 1.25 d. z = 0.40 Draw a vertical line through a normal distribution for each of the following z-score locations. Determine whether the tail is on the right or left side of the line and find the proportion in the body. a. z = 2.50 b. z = 0.80 c. z= 0.50 d. z = 0.77 Find each of the following probabilities for a normal distribution. a. p(z 1.25) b. p(z 0.60) c. p(z 0.70) d. p(z 1.30)What proportion of a normal distribution is located between each of the following z-score boundaries? a. z = 0.25 and z = +0.25 b. z = 0.67 and z = +0.67 c. z = 1.20 and z = +1.20 Find each of the following probabilities for a normal distribution. a. p(0.80 z 0.80) b. p(0.50 z 1.00) c. p(0.20 z 1.50) d. p(1.20 z 0.80)Find the z-score location of a vertical line that separates a normal distribution as described in each of the following. a. 5% in the tail on the left b. 30% in the tail on the right c. 65% in the body on the left d. 80% in the body on the right Find the z-score boundaries that separate a normal distribution as described in each of the following. a. The middle 30% from the 70% in the tails. b. The middle 40% from the 60% in the tails. c. The middle 50% from the 50% in the tails. d.The middle 60% from the 40% in the tails.A normal distribution has a mean of = 70 and a standard deviation of = 8. For each of the following scores, indicate whether the tail is to the right or left of the score and find the proportion of the distribution located in the tail. a. X = 72 b. X= 76 c. X = 66 d. X = 60 A normal distribution has a mean of = 30 and a standard deviation of = 12. For each of the following scores, indicate whether the body is to the right or left of the score and find the proportion of the distribution located in the body. a. X = 32 b. X = 18 c. X = 24 d. X = 39 For a normal distribution with a mean of = 60 and a standard deviation of = 10, find the proportion of the population corresponding to each of the following. a. Scores greater than 65. b. Scores less than 68. c. Scores between 50 and 70.IQ test scores are standardized to produce a normal distribution with a mean of = 100 and a standard deviation of =15. Find the proportion of the population in each of the following IQ categories. a. Genius or near genius: IQ greater than 140 b. Very superior intelligence: IQ between 120 and 140 c. Average or normal intelligence: IQ between 90 and 109 The distribution of SAT scores in normal with = 500 and = 100. a. What SAT score, X value, separates the top 15 % of the distribution from the rest? b. What SAT score, X value, separates the top 10 % of the distribution from the rest? c. What SAT score, X value, separates the top 2 % of the distribution from the rest?According to a recent, people smile an average of = 62 time per day. Assuming that the distribution of smiles is approximately normal with a standard deviation of = 18, find each of the following values. a. What proportion of people smile more than 80 times a day? b. What proportion of people smile at least 50 times a day?A recent newspaper article reported the results of a survey of well-educated suburban parents. The responses to one question indicated that by age 2, children were watching an average of = 60 minutes of television each day. Assuming that the distribution of television-watching times is normal with a standard deviation of = 25 minutes, find each of the following proportions. a. What proportion of 2-year-old children watch more than 90 minutes of television each day? b. What proportion of 2-year-old children watch less than 20 minutes a day?Information from the Department of Motor Vehicles indicates that the average age of licensed drivers is = 45.7 years with a standard deviation of = 12.5 years. Assuming that the distribution of drivers ages is approximately normal, a. What proportion of licensed drivers are older than 50 years old? b. What proportion of licensed drivers are younger than 30 years old?A consumer survey indicates that the average household spends = 185 on groceries each week. The distribution of spending amounts is approximately normal with a standard deviation of = 25. Based on this distribution. a. What proportion of the population spends more than 200 per week on groceries? b. What is the probability of randomly selecting a family that spends less than 150 per week on groceries? c. How much money do you need to spend on groceries each week to be in the top 20% of the distribution A report in 2010 indicates that Americans between the ages of 8 and 18 spend an average of = 7.5 hours per day using some sort of electronic device such as smart phones, computers, or tablets. Assume that the distribution of times is normal with a standard deviation of = 2.5 hours and find the following values. a. What is the probability of selecting an individual who uses electronic devices more than 12 hours a day? b. What proportion of 8- to 18-year-old Americans spend between 5 and 10 hours per day using electronic devices? In symbols. p(5 X 10) = ?Rochester. New York, averages = 21.9 inches of snow for the month of December. The distribution of snowfall amounts is approximately normal with a standard deviation of = 6.5 inches. This year, a local jewelry store is advertising a refund of 50% off of all purchases made in December, if Rochester finishes the month with more than 3 feet (36 inches) of total snowfall. What is the probability that the jewelry store will have to pay off on its promise?A population has a mean of = 65 and a standard deviation of = 16. a. Describe the distribution of sample means (shape, central tendency, and variability) for samples of size n = 4 selected from this population. b. Describe the distribution of sample means (shape, central tendency, and variability) for samples of size n = 64 selected from this population.Describe the relationship between the sample size and the standard error of M.For a population with of = 40 and a standard deviation of = 8, the standard error for a sample mean can never be larger than 8. (True or false.)A sample is selected from a population with a mean of = 90 and a standard deviation of = 15. Find the z-score for the sample mean for each of the following samples. a. n = 9 scores with M = 100 b. n = 25 scores with M = 89 What is the probability of obtaining a sample mean greater than M = 60 for a random sample of n = 16 scores selected from a normal population with a mean of = 65 and a standard deviation of = 20?What are the boundaries for the middle 50% of all possible random samples of n = 25 scores selected from a normal population with = 80 and = 10?If a sample is selected from a population with a mean of = 120 and a standard deviation of = 20, then, on average, how much difference should there be between the sample mean and the population mean a. for a sample of n = 25 scores? b. for a sample of n = 100 scores?Can the value of the standard error ever be larger than the value of population standard deviation? Explain your answer.3LC 4LC A population forms a normal distribution with a mean of . = 80 and a standard deviation of = 20. a. If single score is selected from this population, how much distance would you expect, on average, between the score and the population mean? b. If a sample of n = 100 scores is selected from this population, how much distance would you expect, on average, between the sample mean and the population mean?A population forms a normal shaped distribution with = 40 and = 8. a. A sample of n = 16 scores from this population has a mean of M = 36. Would you describe this as a relatively typical sample, or is the sample mean an extreme value? Explain your answer. b. If the sample from part a had n = 4 scores, would it be considered typical or extreme?The SAT scores for the entering freshman class at a local college form a normal distribution with a mean of = 530 and a standard deviation of = 80. a. For a random sample of n = 16 students, what range of values for the sample mean would be expected 95% of the time? b. What range of values would be expected 95% of the time if the sample size were n = 100?4LC Describe the distribution of sample means (shape, expected value, and standard error) for samples of n = 100 selected from a population with a mean of = 40 and a standard deviation of = 10.A sample is selected from a population with a mean = 40 and a standard deviation of = 8. a. If the sample has n = 4 scores, what is the expected value of M and the standard error of M? b. If the sample has n = 16 scores, what is the expected value of M and the standard error of M?The distribution of sample means is not always a normal distribution. Under what circumstances is the distribution of sample means not normal?A population has a standard deviation of = 24. a. On average, how much difference should exist between the population mean and the sample mean for n = 4 scores randomly selected from the population? b. On average, how much difference should exist for a sample of n = 9 scores? c. On average, how much difference should exist for a sample of n = 16 scores?For a population with a mean of = 70 and a standard deviation of = 20, how much error, on average, would you expect between the sample mean (M) and the population mean for each of the following sample sizes? a. n = 4 scores b. n = 16 scores c. n = 25 scores For a population with a standard deviation of = 20, how large a sample is necessary to have a standard error that is: a. less than or equal to 5 points? b. less than or equal to 2 points? c. less than or equal to 1 point?For a population with = 12, how large a sample is necessary to have a standard error that is: a. less than 4 points? b. less than 3 points? c. less than 2 point?For a sample of n = 25 scores, what is the value of the population standard deviation () necessary to produce each of the following a standard error values? a. M = 10 points? b. M= 5 points? c. M= 2 points?For a population with a mean of = 80 and a standard deviation of = 12, find the z-score corresponding to each of the following samples. a. M = 83 for a sample of n = 4 scores b. M = 83 for a sample of n = 16 scores c. M = 83 for a sample of n = 36 scores A sample of n = 4 scores has a mean of M = 75. Find the z-score for this sample: a. If it was obtained from a population with = 80 and = 10. b. If it was obtained from a population with . = 80 and = 20. c. If it was obtained from a population with . = 80 and = 40.A normal distribution has a mean of = 60 and a standard deviation of = 18. For each of the following samples, compute the z-score for the sample mean and determine whether the sample mean is a typical, representative value or an extreme value for a sample of this size. a. M = 67 for n = 4 scores b.M = 67 for n = 36 scores A random sample is obtained from a normal population with a mean of = 95 and a standard deviation of = 40. The sample mean is M = 86. a. Is this a representative sample mean or an extreme value for a sample of n = 16 scores? b. Is this a representative sample mean or an extreme value for a sample of n = 100 scores?The population of IQ scores forms a normal distribution with a mean of = 100 and a standard deviation of = 15. What is the probability of obtaining a sample mean greater than M = 97. a. for a random sample of n = 9 people? b. for a random sample of n = 25 people?The scores on a standardized mathematics test for 8th-grade children in New York State form a normal distribution with a mean of = 70 and a standard deviation of = 10. a. What proportion of the students in the state have scores less than X = 75? b. If samples of n = 4 are selected from the population, what proportion of the samples will have means less than M = 75? c. If samples of n = 25 are selected from the population, what proportion of the samples will have means less than M = 75?A normal distribution has a mean of = 54 and standard deviation of = 6. What is the probability of randomly selecting a score less then X = 51? What is the probability of selecting a sample of n = 4 scores with a mean less than M = 51? c. What is the probability of selecting a sample of n = 36 scores with a mean less than M 51?A population of scores forms a normal distribution with a mean of = 80 and a standard deviation of = 10. a. What proportion of the scores have values between 75 and 85? b. For samples of n = 4, what proportion of the samples will have means between 75 and 85? c. For samples of n = 16, what proportion of the samples will have means between 75 and 85?For random samples of size n = 25 selected from a normal distribution with a mean of = 50 and a standard deviation of = 20, find each of the following: a. The range of sample means that defines the middle 95% of the distribution of sample means. b. The range of sample means that defines the middle 99% of the distribution of sample means.The distribution ages for students at the state college is positively skewed with a mean of = 21.5 and a standard deviation of = 3. a. What is the probability of selecting a random sample of n = 4 students with an average age greater than 23? (Careful: This is a trick question.) b. What is the probability of selecting a random sample of n = 36 students with an average age greater than 23? c. For a sample of n = 36 students, what is the probability that the average age is between 21 and 22?At the end of the spring semester, the Dean of Students sent a survey to the entire freshman class. One question asked the students how much weight they had gained or lost since the beginning of the school year. The average was a gain of . = 9 pounds with a standard deviation of = 6. The distribution of scores was approximately normal. A sample of n = 4 students is selected and the average weight change is computed for the sample. a. What is the probability that the sample mean will be greater than M = 10 pounds? In symbols, what is p(M 10)? b. Of all of the possible samples, what proportion will show an average weight loss? In symbols, what is p(M 0)? c. What is the probability that the sample mean will be a gain of between M = 9 and M = 12 pounds? In symbols, what is p(9 M 12)?Jumbo shrimp are those that require 10 to 15 shrimp to make a pound. Suppose that the number of jumbo shrimp in a 1-pound bag averages . = 12.5 with a standard deviation of s = 1, and forms a normal distribution. What is the probability of randomly picking a sample of n = 25 1-pound bags that average more than M = 13 shrimp per bag?The average age for licensed drivers in the county is = 40.3 years with a standard deviation of = 13.2 years. a. A researcher obtained a random sample of n = 16 parking tickets and computed an average age of M = 38.9 years for the drivers. Compute the z-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of n = 16 people is a representative sample of licensed drivers? b. The same researcher obtained a random sample of n = 36 speeding tickets and computed an average age of M = 36.2 years for the drivers. Compute the z-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of n = 36 people is a representative sample of licensed drivers?Callahan (2009) conducted a study to evaluate the effectiveness of physical exercise programs for individuals with chronic arthritis. Participants with doctor-diagnosed arthritis either received a Tai Chi course immediately or were placed in a control group to begin the course 8 weeks later. At the end of the 8-week period, self-reports of pain were obtained for both groups. Data similar to the results obtained in the study are shown in the following table. Self-Reported Level of Pain Mean SE Tai Chi course 3.7 1.2 No Tai Chi course 7.6 1.7 a. Construct a bar graph that incorporates all of the information in the table. b. Looking at your graph, do you think that participation in the Tai Chi course reduces arthritis pain?Xu and Garcia (2008) conducted a research study demonstrating that 8-month-old infants appear to recognize which samples are likely to be obtained from a population and which are not. In the study, the infants watched as a sample of n = 5 ping-pong balls was selected from a large box. In one condition, the sample consisted of 1 red ball and 4 white balls. After the sample was selected, the front panel of the box was removed to reveal the contents. In the expected condition, the box contained primarily white balls like the sample, and the infants looked at it for an average of M = 7.5 seconds. In the unexpected condition, the box had primarily red balls, unlike the sample, and the infants looked at it for M = 9.9 seconds. The researchers interpreted the results as demonstrating that the infants found the unexpected result surprising and. therefore, more interesting than the expected result. Assuming that the standard error for both means is M =1 second, draw a bar graph showing the two sample means using brackets to show the size of the standard error for each mean.The city school district is considering increasing class size in the elementary schools. However, some members of the school board are concerned that larger classes may have a negative effect on student learning. In words, what would the null hypothesis say about the effect of class size on student learning?2LC 3LC A researcher selects a sample of n = 16 individuals from a normal population with a mean of = 40 and = 8. A treatment is administered to the sample and after treatment, the sample mean is M = 43. If the researcher uses a hypothesis test to evaluate the treatment effect, what z-score would be obtained for this sample?A small value (near zero) for the z-score statistic is evidence that the sample data are consistent with the null hypothesis. (True or false?)3LCA 1LC 2LC 3LC 4LC If a sample mean is in the critical region with = .01, it would still (always) be in the critical region if alpha were changed to = .05. (True or false?)1LC In a research report, the term significant is used when the null hypothesis is rejected. (True or false?)3LC 4LC 5LC A researcher selects a sample from a population with a mean of . = 60 and administers a treatment to the individuals in the sample. If the researcher predicts that the treatment will increase scores, then a. Using symbols, state the hypotheses for a one-tailed test. b. For the one-tailed test, would the critical region be located in the right-hand tail or the left-hand tail of the distribution?2LC A researcher obtains z = 2.43 for a hypothesis lest. Using = .01, the researcher should reject the null hypothesis for a one-tailed test hut fail to reject for a two-tailed test. (True or false?)1LC A researcher selects a sample from a population with . = 45 and = 8. A treatment is administered to the sample and. after treatment, the sample mean is found to be M = 47. Compute Cohens d to measure the size of the treatment effect.3LC 1LC For a 5-point treatment effect, a researcher computes power of p = 0.50 for a two-tailed hypothesis test with = .05. a. Will the power increase or decrease for a 10-point treatment effect? b. Will the power increase or decrease if alpha is changed to = .01? c. Will the power increase or decrease if the researcher changes to a one-tailed test?3LC 4LC The value of the z-score in a hypothesis test is influenced by a variety of factors. Assuming that all other variables are held constant, explain how the value of z is influenced by each of the following: a. An increase in the difference between the sample mean and the original population mean. b. An increase in the population standard deviation. c. An increase in the number of scores in the sample.Define the alpha level and the critical region, and explain how they are related.Although there is a popular belief that herbal remedies such as Ginkgo biloba and Ginseng may improve learning and memory in healthy adults, these effects are usually not supported by well-controlled research (Person, Bringlov, Nilsson, Nyberg, 2004). In a typical study, a researcher obtains a sample of n = 16 participants and has each person take the herbal supplements every day for 90 days. At the end of the 90 days, each person takes a standardized memory test. For the general population, scores from the test form a normal distribution with a mean of = 50 and a standard deviation of = 12. The sample of research participants had an average of M = 54. a. Assuming a two-tailed test, state the null hypothesis in a sentence that includes the two variables being examined. b. Using the standard 4-step procedure, conduct a two-tailed hypothesis test with = .05 to evaluate the effect of the supplements.Childhood participation in sports, cultural groups, and youth groups appears to be related to improved selfesteem for adolescents (McGee, Williams, Howden-Chapman, Martin, Karachi, 2006). In a representative study, a sample of n 100 adolescents with a history of group participation is given a standardized self-esteem questionnaire. For the general population of adolescents, scores on this questionnaire form a normal distribution with a mean of 50 and a standard deviation of = 15 The sample of group participation adolescents had an average of M = 53.8. a. Does this sample provide enough evidence to conclude that self-eastern scores for these adolescents are significantly different from those of the general population? Use a two-tailed test with .05. b. Compute Cohens d to measure the size of the difference. c. Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.A local college requires an English composition course for all freshmen. This year they are evaluating a new online version of the course. A random sample of n = 16 freshmen is selected and the students are placed in the online course. At the end of the semester, all freshmen take the same English composition exam. The average score for the sample is M = 76. For the general population of freshmen who look the traditional lecture class, the exam scores form a normal distribution with a mean of = 80. a. If the final exam scores for the population have a standard deviation of = 12, does the sample provide enough evidence to conclude that the new online course is significantly different from the traditional class? Assume a two-tailed test with = .05. c. If the population standard deviation = 6, is the sample sufficient to demonstrate a significant difference? Again, assume a two-tailed test with = .05. c. Comparing your answers for pans a and b, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test.A random simple is selected from a normal population with a mean of = 30 and a standard deviation of = 8. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 33. a. If the sample consists of n = 16 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with = .05. b. If the sample consists of n = 64 scores, is the sample mean sufficient effect? Use a two-tailed test with = .05. c. Comparing your answers for parts a and b, explain how the size of the sample influences the outcome of a hypothesis test.A random sample of n = 25 scores is selected from a normal population with a mean of = 40. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 44. a. If the population standard deviation is = 5, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with = .05. b. If the population standard deviation is = 15, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with = .05. c. Comparing your answers for parts a and b, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test.Brunt, Rhee, and Zhong (2008) surveyed 557 undergraduate college students to examine their weight status, health behaviors, and diet. Using body mass index (BMI), they classified the students into four categories; underweight, healthy weight, overweight, and obese. They also measured dietary variety by counting the number of different foods each student ate from several food groups. Note that the researchers are not measuring the amount of food eaten, but rather the number of different foods eaten (variety, not quantity). Nonetheless, it was somewhat surprising that the results showed no differences among the tour weight categories that were related to eating fatty and/or sugary snacks. Suppose a researcher conducting a follow up study obtains a sample of n = 25 students classified as healthy weight and a sample of n = 36 students classified as overweight Each student completes the food variety questionnaire, and the healthy-weight group produces a mean of M = 4.01 for the fatty, sugary snack category compared to a mean of M = 4.48 for the overweight group. The results from the Brunt, Rhee, and Zhong study showed an overall mean variety score of = 4.22 for the discretionary sweets or fats food group. Assume that the distribution of scores is approximately normal with a standard deviation of = 0.60. a. Does the sample of n = 36 indicate that the number of fatty, sugary snacks eaten by overweight students is significantly different front the overall population mean? Use a two-tailed test with = .05. b. Based on the sample of n = 25 healthy-weight students, can you conclude that healthy-weight students eat significantly fewer fatty, sugary snacks than the overall population? Use a one-tailed test with = .05.A random simple is selected from a normal population with a mean of = 100 and a standard deviation of = 20. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 96. a. How large a sample is necessary for this sample mean to be statistically significant? Assume a two-tailed test with = .05. b. If the sample mean were M = 98, what sample size would be needed to be significant for a two-tailed test with = .05?In a study examining the effect of alcohol on reaction time, Liguori and Robinson (2001) found that even moderate alcohol consumption significantly slowed response time to an emergency situation in a driving simulation, In a similar study, researchers measured reaction time 30 minutes alter participants consumed one 6-ounce glass of wine. Again, they used a standardized driving simulation task for which the regular population averages = 400 msec. The distribution of reaction times is approximately normal with = 40. Assume that the researcher obtained a sample mean of M = 422 for the n = 25 participants in the study. a. Are the data sufficient to conclude that the alcohol has a significant effect on reaction time? Use a two-tailed test with = .01. b. Do the data provide evidence that the alcohol significantly increased (slowed) reaction time? Use a one-tailed lest with = .05. c. Compute Cohen's d to estimate the size of the effect.The researchers cited in the previous problem (Liguori Robinson. 2001) also examined the effect of caffeine on response time in the driving simulator. In a similar study, researchers measured reaction time 30 minutes after participants consumed one 6-ounce cup of coffee. Using the same driving simulation task, for which the distribution of reaction times is normal with = 400 msec, and = 40, they obtained a mean of M = 392 for a sample of n = 36 participants. a. Are the data sufficient to conclude that caffeine has a significant effect on reaction time? Use a two-tailed test with = .05. b. Compute Cohens d to estimate the size of the effect. c. Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.There is some evidence indicating that people with visible tattoos ore viewed more negatively than people without visible tattoos (Resenhoeft, Villa, Wiseman. 2008). In a similar study, a researcher first obtained overall ratings of attractiveness for a woman with no tattoos shown in a color photograph, On a 7-point scale, the woman received an average rating of = 4.9, and the distribution of ratings was normal with a standard deviation of = 0.84. The researcher then modified the photo by adding a tattoo of a butterfly on the woman's left arm. The modified photo was then shown to a sample of n = 16 students at a local community college and the students used the same 7-point scale to rate the attractiveness of the woman the average score for the photo with the tattoo was M = 4.2. a. Do the data indicate a significant difference in rated attractiveness when the woman appeared to have a tattoo? Use a two-tailed test with = .05 b. Compute Cohen's d to measure the size of the effect. c. Write a sentence describing the outcome of the hypothesis test and the measure of effect sire as it would appear in a research report.

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