Exam 6: Sampling Distributions

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 median for a random sample of $n = 2$ observations from this population. c. Show that the median is an unbiased estimator of $\mu$ .

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

If $\bar { x }$ is a good estimator for ?, then we expect the values of $\bar { x }$ to cluster around ?.

One year, the distribution of salaries for professional sports players had mean $1.8 million and standard deviation $0.6 million. Suppose a sample of 400 major league players was taken. Find the approximate probability that the average salary of the 400 players that year exceeded $1.1 million.

The probability distribution shown below describes a population of measurements x 0 3 6 9 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 $\bar { x }$ . histogram for the sampling distribution of x.

The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, μ and σ, are statistics.

The length of time a traffic signal stays green (nicknamed the "green time") at a particular intersection follows a normal probability distribution with a mean of 200 seconds and the standard deviation of 10 seconds. Use this information to answer the following questions. Which of the following describes the derivation of the sampling distribution of the sample mean?

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.

A point estimator of a population parameter is a rule or formula which tells us how to use sample data to calculate a single number that can be used as an estimate of the population parameter.

The Central Limit Theorem is important in statistics because _____.

The weight of corn chips dispensed into a 15-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 15.5 ounces and a standard deviation of 0.1 ounce. Suppose 400 bags of chips are randomly selected. Find the probability that the mean weight of these 400 bags exceeds 15.6 ounces.

The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following is a necessary condition for the Central Limit Theorem to be used?

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

Consider the probability distribution shown here. x 1 3 5 p(x) Let be the sample mean for random samples of n=2 measurements from this distribution. Find E(x) and E() . 6.2 Properties of Sampling Distributions: Unbiasedness and Minimum Variance 1 Understand Unbiasedness

Which of the following describes what the property of unbiasedness means?

Suppose a random sample of n = 64 measurements is selected from a population with mean μ = 65 and standard deviation $\sigma = 12 \text {. Find the probability that } \bar { x } \text { falls between } 65.75 \text { an } 68.75 \text {. }$

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? A) 0 2 4 () 1/3 1/3 1/3 B) 0 1 2 3 4 () 1/5 1/5 1/5 1/5 1/5 C) 0 1 2 3 4 () 2/9 2/9 1/9 2/9 2/9 D) 0 1 2 3 4 () 1/9 2/9 3/9 2/9 1/9

The Central Limit Theorem is considered powerful in statistics because __________.

Suppose a random sample of n = 64 measurements is selected from a population with mean μ = 65 and standard deviation $\sigma = 12 \text {. Find the } z \text {-score corresponding to a value of } \bar { x } = 68 \text {. }$ 3 Find Probability

The average score of all golfers for a particular course has a mean of 73 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 74.