A variable of a population is normally distributed with mean µ and standard deviation σ. For samples of size n, fill in the blanks. Justify your answers. a. Approximately 68% of all possible samples have means that lie within of the population mean, µ.b. Approximately 95% of all possible samples have means that lie within of the population mean, µ.c. Approximately 99.7% of all possible samples have means that lie within of the population mean, µ.d. 100(1 - a)% of all possible samples have means that lie within of the population mean, µ. (Hint: Draw a graph for the distribution of x, and determine the z-scores dividing the area under the normal curve into a middle 1 - a area and two outside areas of a/2.)

Question

A variable of a population is normally distributed with mean µ and standard deviation σ. For samples of size n, fill in the blanks. Justify your answers.

a. Approximately 68% of all possible samples have means that lie within of the population mean, µ.
b. Approximately 95% of all possible samples have means that lie within of the population mean, µ.
c. Approximately 99.7% of all possible samples have means that lie within of the population mean, µ.
d. 100(1 - a)% of all possible samples have means that lie within of the population mean, µ. (Hint: Draw a graph for the distribution of x, and determine the z-scores dividing the area under the normal curve into a middle 1 - a area and two outside areas of a/2.)

Expert Answer

Want to see the step-by-step answer?

Check out a sample Q&A here.

Want to see this answer and more?

Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes!*

*Response times may vary by subject and question complexity. Median response time is 34 minutes for paid subscribers and may be longer for promotional offers.
Tagged in
Math
Statistics

Related Statistics Q&A

Find answers to questions asked by students like you.

Q: Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothe...

A: The given claim is that subjects who respond in favor is equal to 0.5.The null and the alternative h...

Q: Social Networking In a Pew Research Center survey of Internet users, 3732 respondents say that they ...

A: The total number of respondents is,

Q: A variable has the density curve whose equation is y = 2x for 0< x< 1, and y = 0 otherwise. a....

A: Click to see the answer

Q: A random variable X equals 2 with probability 0.386. a. Use probability notation to express that fac...

A: (a)Justification:The probability notation is P(X=2)=0.386.(b)Justification:To obtain the percentage ...

Q: In each of Exercises, assume that the variable under consideration has a density curve. Note that th...

A: a.It is given that 28.4% of all possible observations of the variable are less than 11.The probabili...

Q: Hurricanes. In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean num...

A: a)Define the random variable x as the number of hurricanes in a year. Here, the number of hurricanes...

Q: In each of Exercises, assume that the variable under consideration has a density curve. Note that th...

A: a)Here, it is observed that the percentage of all possible observation of the variable that lies wit...

Q: examples of graphic presentation of data

A: Line graphs:Graphs are used to represent statistical data. A line graph is a diagram showing the rel...

Q: The following data represents the age of 30 lottery winners.   24 29 31 32 34 34 38 40 41 42 44...

A: Click to see the answer

Q: Fire Loss. The loss, in millions of dollars, due to a fire in a commercial building is a variable wi...

A: (a)The density curve of the variable is,

Q: A local store is having a sale. All shoes are 40% off. If a pair of shoes normally costs $13.72 plus...

A: Note:Hey there! Thank you for posting the question. Since exact value is not given, from the given i...

Q: Why should you generally expect some error when estimating a parameter (e.g., a population mean) by ...

A: Solution: We have to determine why should we generally expect some error when estimating a parameter...

Q: A variable of a population has a left-skewed distribution. a. If a large simple random sample is tak...

A: a.The information represents the two simple random samples are taken from the population and the pop...

Q: What does it mean for two or more events to be mutually exclusive?

A: Mutually exclusive eventsTwo events are said to be mutually exclusive if they have no element in com...

Q: Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothe...

A: Null and alternative hypotheses:The null and alternative hypotheses are as follows:H0: p = 0.10H1: p...

Q: In each of Problems, assume that the variable under consideration has a density curve. Note that the...

A: The percentage of all possible observation of the variable that lie within any specified range equal...

Q: Under what circumstances is using a normal probability plot to assess the normality of a variable us...

A: Here, it is observed that the normal probability plot determines the shape of the distribution bette...

Q: Notation When randomly selecting an adult, A denotes the event of selecting someone with blue eyes. ...

A: Let the event A denote the randomly selected adult with blue eyes.Probability:The probability of hap...

Q: A variable is normally distributed with mean 6 and standard deviation 2. a. Determine and interpret ...

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If y...

Q: Significance with Range Rule of Thumb. In Exercises 31 and 32, assume that hybridization experiments...

A: a.Mean:The mean number of peas with green pods in the groups of 16 is,

Q: Explain in detail what a normal probability plot is and how it is used to assess the normality of a ...

A: Normal probability plot:The plot for the observed values of a variable versus the normal scores is r...

Q: Precision is a measure of -?-?

A: Precision:In measurement of a set, accuracy refers to closeness of the measurements to a specific va...

Q: A and B are events such that P(A) = 0.4,P(B) = 0.5, and P(A&B) = 0.2. Answer each question and e...

A: a.Mutually Exclusive Events:Two events are said to be mutually exclusive if they cannot occur at the...

Q: 10. Who established: 3 <T < 3 70 11. What is log1o 5 to 3 sig. figs.? 12. What is log 3 to 2 sig. fi...

A: Hey there! Thank you for posting the question. Since there are multiple questions posted, according ...

Q: In Exercises 1–5, assume that 74% of randomly selected adults have a credit card (based on results f...

A: Define the random variable x as the number of adults having credit card. Here, the number of adults ...

Q: *Sweet Sixteen. In the NCAA basketball tournament, 64 teams compete in 63 games during six rounds of...

A: The formula for combination is given below:

Q: Standard Normal Distribution Identify the requirements necessary for a normal distribution to be a s...

A: The requirements necessary for a normal distribution to be a standard normal distribution are,

Q: 9. True or False? Continuous variables, sometimes called measurement or quantitative variables, take...

A: Click to see the answer

Q: Births in the United States In the United States, the true probability of a baby being a boy is 0.51...

A: Given:Probability of baby being a boy = 0.512Number of births selected n = 5We have to find probabil...

Q: What does the probability distribution of a discrete random variable tell you?

A: The discrete random variable is obtained by counting values for which there are no in-between values...

Q: Earthquakes. According to The Earth: Structure, Composition and Evolution (The Open University, S237...

A: a)Mean:The sum of all the entries divided by the total number of entries is known mean.The formula f...

Q: decimal equivalent to 3/2pi

A: Note:Hey there! Thank you for posting the question. For the calculation we are considering given val...

Q: In your biology class, your final grade is based on several things: a lab score, scores on two major...

A: The data is provided for the final grades of a biology class.Here, the x values are scores 93, 65, 7...

Q: In Exercises 25–28, find the probabilities and answer the questions. Too Young to Tat Based on a Har...

A: a)Define the random variable x as the number of adults selected who regret getting tattoos saying th...

Q: Let 0&lt; a&lt; 1. For a normally distributed variable, show that 100(1 - a)% of all possible observ...

A: Click to see the answer

Q: Mega Millions As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot...

A: Solution: We are given that:Winning the jackpot requires that you select the correct five different ...

Q: Four actions are to be performed in a definite order. There are three possibilities for the first ac...

A: As per the information given in the questionTotal no of actions = 4Action 1 – 3 possibilitiesAction ...

Q: Fill in the blanks. a. A ______ is a quantitative variable whose value depends on chance.b. A discre...

A: Click to see the answer

Q: Washing Hands Based on results from a Bradley Corporation poll, assume that 70% of adults always was...

A: (a)Define the random variable x as the number of adults washing their hands after using the public r...

Q: Powerball: Poisson Approximation to Binomial There is a 1/292,201,338 probability of winning the Pow...

A: Requirements for using the Poisson distribution as an approximation to binomial distribution are as ...

Q: Can you help me solve and understand these questions?

A: Hey there! Thank you for posting the question. Since your question has more than 3 parts, we are sol...

Q: Change for a Quarter How many different ways can you make change for a quarter? (Different arrangeme...

A: The number of coins that are less than a quarter in terms of money is three. They arePenny ($0.01)Ni...

Q: Medical Helicopters In a study of helicopter usage and patient survival, results were obtained from ...

A: Solution:  We are given that:In a study of helicopter usage and patient survival, results were obtai...

Q: A and B are independent events such that P(A) = 0.3 and P(B) = 0.6. Find P(A&amp;B).

A: Click to see the answer

Q: A data set with roughly a bell-shaped distribution has mean 45 and standard deviation 12. Approximat...

A: The percentage of the observations lie between 33 and 57 is obtained below:Let X be an observation f...

Q: Express the quartiles, Q1, Q2, and Q3, of a normally distributed variable in terms of its mean, µ, a...

A: First Quartile (Q1)From the given information the first quartile is represented by 25th percentile a...

Q: For a normally distributed variable, what is the relationship between the percentage of all possible...

A: Justification:The area under the associated normal curve to the right of 7 is given.From the given i...

Q: Sample for a Poll There are 15,524,971 adults in Florida. If The Gallup organization randomly select...

A:  The selections independent or dependent is obtained below: It is given that there are 15,524,971 ad...