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Exam 6: Sampling Distributions

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Question 31

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

A) it works for any sample size provided the population is normal
B) it works for any sample provided the population distribution is known
C) it works for any population distribution provided the population mean is known
D) it works for any population distribution provided the sample size is sufficiently large

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Question 32

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A random sample of $\mathrm { n } = 400$ 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, $\hat { p }$ .

A) $.21 ; .008$
B) .79; .02
C) .79; .008
D) .21; .02

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Question 33

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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 15 minutes and a standard deviation of $1.4$ minutes. If times were collected from 60 randomly selected walks, describe the sampling distribution of $\bar { x }$ , the sample mean time.

Consider the population described by the probability distribution below. $\begin{array}{c|c|c|c}x & 2 & 5 & 7 \\\hline p(x) & .2 & .5 & .3\end{array}$ The random variable $x$ is observed twice. The observations are independent. The different samples of size 2 and their probabilities are shown below. Find the sampling distribution of the sample mean $\bar { x }$

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 $\mu$ . 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 $\mu$ .

The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $3450 and the standard deviation is $300. 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?

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

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 $\sigma _ { \bar { x } }$ .

A random sample of $\mathrm { n } = 300$ measurements is drawn from a binomial population with probability of success .26. Give the mean and the standard deviation of the sampling distribution of the sample proportion, $\hat { p }$ .

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.

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

Consider the probability distribution shown here. $\begin{array}{l|lll}\hline x & 7 & 9 & 11 \\\hline p(x) & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\\hline\end{array}$ Let $\bar { x }$ be the sample mean for random samples of $n = 2$ measurements from this distribution. Find $E ( x )$ and $E ( \bar { x } )$ .