Exam 5: Sampling Distributions

A random sample of n = 400 measurements is drawn from a binomial population with probability of success .21. Give the mean and the standard deviation of the sampling distribution of the sample Proportion, $\hat { \mathrm { p } }$

The term statistic refers to a population quantity, and the term parameter refers to a sample quantity.

The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.

One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.7 million. Suppose a sample of 100 major league players was taken. Find the approximate probability that the average salary of the 100 players that year exceeded $1.1 million.

The probability distribution shown below describes a population of measurements. 0 2 4 () 1/3 1/3 1/3 Suppose that we took repeated random samples of $n = 2$ observations from the population described above. Find the expected value of the sampling distribution of the sample mean.

Consider the population described by the probability distribution below. x 3 5 7 p(x) .1 .7 .2 a. Find $\mu$ . b. Find the sampling distribution of the sample mean $\bar { x }$ for a random sample of $n = 2$ measurements from the distribution. c. Show that $\bar { x }$ is an unbiased estimator of $\mu$ .

Consider the population described by the probability distribution below. x 0 2 4 p(x) a. Find $\mu$ . b. Find the sampling distribution of the sample mean for a random sample of $n = 3$ measurements from this distribution. c. Find the sampling distribution of the sample median for a random sample of $n = 3$ observations from this population. d. Show that both the mean and the median are unbiased estimators of $\mu$ for this population. e. Find the variances of the sampling distributions of the sample mean and the sample median. f. Which estimator would you use to estimate $\mu$ ? Why?

The probability distribution shown below describes a population of measurements. 0 2 4 () 1/3 1/3 1/3 Suppose that we took repeated random samples of $n = 2$ observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

The probability distribution shown below describes a population of measurements that can assume values of 2, 5, 8, and 11, each of which occurs with the same frequency: x 2 5 8 11 p(x) Consider taking samples of $n = 2$ measurements and calculating $\bar { x }$ for each sample. Construct the probability histogram for the sampling distribution of $\bar { x }$ .

Suppose a random sample of $n = 64$ measurements is selected from a population with mean $\mu = 65$ and standard deviation $\sigma = 12$ . Find the values of $\mu _ { \bar { x } } ^ { - }$ and $\sigma _ { \bar { x } } ^ { - }$ .

Which of the following does the Central Limit Theorem allow us to disregard when working with the sampling distribution of the sample mean?

A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended to estimate.

The average score of all golfers for a particular course has a mean of 66 and a standard deviation of 3.5. Suppose 49 golfers played the course today. Find the probability that the average score of the 49 golfers exceeded 67.

In most situations, the true mean and standard deviation are unknown quantities that have to be estimated.

Suppose a random sample of $n = 64$ measurements is selected from a population with mean $\mu = 65$ and standard deviation $\sigma = 12$ . Find the $z$ -score corresponding to a value of $\bar { x }$ $= 68$ .

Sample statistics are random variables, because different samples can lead to different values of the sample statistics.

A random sample of size n is to be drawn from a population with $\mu = 700 \text { and } \sigma = 200$ What size sample would be necessary in order to reduce the standard error to 10?

The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators.

Which of the following describes what the property of minimum variance means?

Consider the population described by the probability distribution below. x 2 5 7 p(x) .2 .5 .3 The random variable $x$ is observed twice. The observations are independent. The different samples of size 2 and their probabilities are shown below. Sample Probability 2,2 .04 2,5 .10 2,7 .06 Sample Probability 5,2 .10 5,5 .25 5,7 .15 Sample Probability 7,2 .06 7,5 .15 7,7 .09 Find the sampling distribution of the sample mean $\bar { x }$ .