Additional_Exercises

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Economics

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Apr 3, 2024

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Additional Exercises 1. The General Social Survey asked the question, “After an average workday, about how many hours do you have to relax or pursue activities that you enjoy?” to a random sample of 1,155 Americans. The average relaxing time was found to be 1.65 hours. Determine which of the following is an observation, a variable, a sample statistic (value calculated based on the observed sample), or a population parameter. (a) Average number of hours all Americans spend relaxing after an average workday. (b) Number of hours spent relaxing after an average workday. (c) An American in the sample. (d) 1.65. 2. Researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution lev- els were measured by air quality monitoring stations. Specifically, levels of car- bon monoxide were recorded in parts per million, nitrogen dioxide and ozone in parts per hundred million, and coarse particulate matter (PM 10 ) in µg/m 3 . Length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. The analysis suggested that increased ambient PM 10 and, to a lesser de- gree, CO concentrations may be associated with the occurrence of preterm births. (https://www.jstor.org/stable/3703990) (a) Who are the subjects in this study, and how many are included? (b) Comment on whether the results of the study can be generalized to the pop- ulation, and if the findings of the study can be used to establish causal rela- tionships. 1
3. Researchers studying the effect of antibiotic treatment for acute sinusitis com- pared to symptomatic treatments randomly assigned 89 adults diagnosed with acute sinusitis to one of two groups: treatment or control. Study participants received either a 10-day course of amoxicillin (an antibiotic) or a placebo simi- lar in appearance and taste. The placebo consisted of symptomatic treatments such as acetaminophen, nasal decongestants, etc. At the end of the 10-day pe- riod, patients were asked if they experienced improvement in symptoms. The distribution of responses is summarized below. (https://jamanetwork.com/journals/jama/fullarticle/1104985) (a) What percent of patients in the treatment group experienced improvement in symptoms? (b) What percent experienced improvement in symptoms in the control group? (c) In which group did a higher percentage of patients experience improvement in symptoms? (d) Your findings so far might suggest a real difference in the effectiveness of an- tibiotic and placebo treatments for improving symptoms of sinusitis. How- ever, this is not the only possible conclusion. What is one other possible ex- planation for the observed difference between the percentages patients who experienced improvement in symptoms? (e) What are the explanatory and response variables in this study? 4. In a study of the relationship between socio-economic class and unethical be- havior, 129 University of California undergraduates at Berkeley were asked to identify themselves as having low or high social class by comparing themselves to others with the most (least) money, most (least) education, and most (least) respected jobs. They were also presented with a jar of individually wrapped candies and informed that the candies were for children in a nearby laboratory, but that they could take some if they wanted. After completing some unre- lated tasks, participants reported the number of candies they had taken. It was found that those who were identified as upper-class took more candy than others. (https://www.pnas.org/content/109/11/4086) 2
(a) Identify the population of interest and the sample in this study. (b) Comment on whether the results of the study can be generalized to the pop- ulation, and if the findings of the study can be used to establish causal rela- tionships. 5. Researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution lev- els were measured by air quality monitoring stations. Specifically, levels of car- bon monoxide were recorded in parts per million, nitrogen dioxide and ozone in parts per hundred million, and coarse particulate matter (PM 10 ) in µg/m 3 . Length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. The analysis suggested that increased ambient PM 10 and, to a lesser de- gree, CO concentrations may be associated with the occurrence of preterm births. (https://www.jstor.org/stable/3703990) (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) What are the variables in the study? Identify each variable as numerical or categorical. If numerical, state whether the variable is discrete or continuous. If categorical, state whether the variable is ordinal. 6. The bar graph and the pie chart below show the distribution of pre-existing medi- cal conditions of children involved in a study on the optimal duration of antibiotic use in treatment of tracheitis, which is an upper respiratory infection. (a) What features are apparent in the bar graph but not in the pie chart? (b) What features are apparent in the pie chart but not in the bar graph? (c) Which graph would you prefer to use for displaying these categorical data? 3
7. A local news survey asked 500 randomly sampled Los Angeles residents which shipping carrier they prefer to use for shipping holiday gifts. The table below shows the distribution of responses by age group as well as the expected counts for each cell. (a) Which graph (top or bottom) would you use to understand the shipping choices of people of different ages? (b) Which graph (top or bottom) would you use to understand the age distribu- tion across different types of shipping choices? (c) A new shipping company would like to market to people over the age of 55. Who will be their biggest competitor? (d) FedEx would like to reach out to grow their market share to balance the age demographics of FedEx users. To what age group should FedEx market? 8. Here are the starting salaries, in thousands of dollars, offered to the 20 students who earned degrees in computer science in 2011 at a university. 63 56 66 77 50 53 78 55 90 65 64 69 59 76 48 54 49 68 51 50 (a) Make a graph to describe the distribution and write a brief description of its important features. (b) Find the median salary. (c) Find the mean salary. (d) Find the mode of the salaries. 4
(e) Is the mean about the same as the median or not? What feature of the dis- tribution explains the difference between the mean and the median? Is the mode a good measure of the center for these data? 9. Each month, the Commerce Department reports the “average” price of new single- family homes. For August 2012, the two “averages” reported were $256,900 and $295,300. Which of these numbers was the mean price and which was the median price? Explain your answer. 10. In 1961 New York Yankee outfielder Roger Maris held the major league record for home runs in a single season, with 61 home runs. That record held for 37 years. Here are Maris’s home run totals for his 10 years in the American League. 13 23 26 16 33 61 28 39 14 8 (a) Find the mean number of home runs that Maris hit in a year, both with and without his record 61. How does removing the record number of home runs affect his mean number of runs? (b) Find the median number of home runs that Maris hit in a year, both with and without his record 61. How does removing the record number of home runs affect his median number of runs? (c) If you had to choose between the mean and median to describe Maris’s home run hitting pattern, which would you use? 11. In a class of 25 students, 24 of them took an exam in class and 1 student took a make-up exam the following day. The professor graded the first batch of 24 exams and found an average score of 74 points with a standard deviation of 8.9 points. The student who took the make-up the following day scored 64 points on the exam. (a) Does the new student’s score increase or decrease the average score? (b) What is the new average? (c) Does the new student’s score increase or decrease the standard deviation of the scores? 12. Compare the two plots below. What characteristics of the distribution are appar- ent in the histogram and not in the box plot? What characteristics are apparent in the box plot but not in the histogram? 5
13. Estimate the median for the 400 observations shown in the histogram, and note whether you expect the mean to be higher or lower than the median. 14. For each of the following, state whether you expect the distribution to be sym- metric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR. Explain your reasoning. (a) Housing prices in a country where 25% of the houses cost below $350,000, 50% of the houses cost below $450,000, 75% of the houses cost below $1,000,000, and there are a meaningful number of houses that cost more than $6,000,000. (b) Housing prices in a country where 25% of the houses cost below $300,000, 50% of the houses cost below $600,000, 75% of the houses cost below $900,000, and very few houses that cost more than 1,200,000. (c) Number of alcoholic drinks consumed by college students in a given week. Assume that most of these students don’t drink since they are under 21 years old, and only a few drink excessively. 6
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