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Exam 6: Sampling Distributions

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Question 21

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

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Question 22

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

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The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 14 minutes and a standard deviation of 1.1 minutes. If times were collected from 40 randomly selected walks, describe the sampling distribution of $\bar{x}$ , the sample mean time.

Consider the population described by the probability distribution below. $\begin{array} { c | c | c | c } x & 0 & 2 & 4 \\\hline p ( x ) & \frac { 1 } { 3 } & \frac { 1 } { 3 } & \frac { 1 } { 3 }\end{array}$ a. Find μ. 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 μ 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 μ? Why? 6.3 The Sampling Distribution of x-bar and the Central Limit Theorem 1 Understand Central Limit Theorem

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: $\begin{array}{l|rrrr}\hline x & 2 & 5 & 8 & 11 \\\hline p(x) & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\\hline\end{array}$ Find $E ( x ) = \mu$ . Then consider taking samples of $n = 2$ measurements and calculating $\bar { x }$ for each sample. Find the expected value, $E ( \bar { x } )$ , of $\bar { x }$ .

The sampling distribution of the sample mean is shown below. $\begin{array} { c | c c c c c } \overline { \mathrm { x } } & 4 & 5 & 6 & 7 & 8 \\\hline \mathrm { p } ( \overline { \mathrm { x } } ) & 1 / 9 & 2 / 9 & 3 / 9 & 2 / 9 & 1 / 9\end{array}$ Find the expected value of the sampling distribution of the sample mean.

The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of .2 ounce. Suppose 100 bags of chips are randomly selected. Find the probability that the mean weight of these 100 bags exceeds 10.45 ounces.

The ideal estimator has the greatest variance among all unbiased estimators.

The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently large.

The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $2700 and the standard deviation is $400. The distribution is skewed to the right due to several high volume days (football game days) . Suppose that 100 days are randomly selected and the average daily revenue computed. Which of the following describes the sampling distribution of the sample mean?

As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well.

Which of the following statements about the sampling distribution of the sample mean is incorrect?

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

The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 4.2 cars and a standard deviation of 6. The number of cars running the red light was observed on 100 randomly chosen days and the mean number of cars calculated. Describe the sampling distribution of the sample mean.

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

The sample mean, $\bar { x }$ , is a statistic.

When estimating the population mean, the sample mean is always a better estimate than the sample median.

A random sample of size n is to be drawn from a population with μ = 1500 and σ = 200. What size sample would be necessary in order to reduce the standard error to 20? 2 Find Mean, Standard Deviation

Consider the population described by the probability distribution below. $\begin{array}{c|c|c|c}x & 3 & 5 & 7 \\\hline p(x) & .1 & .7 & .2\end{array}$ a. Find $\sigma ^ { 2 }$ . b. Find the sampling distribution of the sample variance $s ^ { 2 }$ for a random sample of $n = 2$ measurements from the distribution. c. Show that $s ^ { 2 }$ is an unbiased estimator of $\sigma ^ { 2 }$ .

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