Deck 8: Sampling Distributions

Question 1

The mean of the sampling distribution of the sample proportion , when the sample size n = 100 and the population proportion p = 0.92, is 92.0.

Answer 1

False

Question 2

Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals 2.50.

Answer 2

True

Question 3

The amount of time it takes to complete a final examination is negatively skewed distribution with a mean of 70 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 73.5 minutes is approximately 0.

Answer 3

True

Question 4

If all possible samples of size n are drawn from a normal population, the probability distribution of the sample mean is an exact normal distribution.

Question 5

As the size of the sample is increased, the standard error of decreases.

Question 6

A sample of size 25 is selected from a population of size 75. The finite population correction needed to find the standard error of .

Question 7

A sampling distribution is a probability distribution for a statistic, not a parameter.

Question 8

If the population distribution is skewed, in most cases the sampling distribution of the sample mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Question 9

If the population distribution is unknown, in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Question 10

A sample of size n is selected at random from an infinite population. As n increases, the standard error of the sample mean increases.

Question 11

If all possible samples of size n are drawn from an infinite population with standard deviation 8, then the standard error of the sample mean equals 1.0 if the sample size is 64.

Question 12

If the sample size increases, the standard error of the mean also increases.

Question 13

A sample of size 25 is selected from a population of size 500. The finite population correction is needed to find the standard error of .

Question 14

When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample mean is normal regardless of the sample size n.

Question 15

As the size of the sample is increased, the mean of increases.

Question 16

The Central Limit Theorem permits us to draw conclusions about a population based on a sample alone, without having any knowledge about the distribution of that population. And this works no matter what the sample size is.

Question 17

The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces. Suppose we take a random sample of 36 bottles filled by this machine. The sampling distribution of the sample mean has a standard error of 0.25 ounces.

Question 18

A sampling distribution is defined as the probability distribution of means from all possible sample sizes that are taken from a given population.

Question 19

The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces. We take a random sample of 36 bottles filled by this machine. The sampling distribution of the sample mean has a mean of 50 ounces.

Question 20

In inferential statistics, the standard error of the sample mean assesses the uncertainty or error of estimation.

Question 21

The Central Limit Theorem states that, if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean :

A) is approximately normal if n < 30.
B) is approximately normal if n > 30.
C) is approximately normal if the underlying population is normal.
D) None of these choices.

Question 22

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if the sample size n is greater than or equal to 30.

Question 23

In general, the binomial probability P(X = x) is approximated by the area under a normal curve between x- .5 and x + .5.

Question 24

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, is less than 78 is:

A) 0.9332
B) 0.5987
C) 1.5000
D) None of these choices.

Question 25

Recall the rule of thumb used to indicate when the normal distribution is a good approximation of the sampling distribution for the sample proportion . For the combination n = 50, p = 0.05, the rule is satisfied.

Question 26

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

A) sample size is smaller than 5% of the population size.
B) sample size is greater than 5% of the sample size.
C) sample size is more than half of the population size.
D) None of these choices.

Question 27

Random samples of size 49 are taken from an infinite population whose mean is 300 and standard deviation is 21. The mean and standard error of the sample mean, respectively, are:

A) 300 and 21
B) 300 and 3
C) 300 and 0.43
D) None of these choices.

Question 28

A sample of size 25 is selected at random from a finite population. If the finite population correction factor is 0.63, then the population size is:

A) 25
B) 66
C) 41
D) None of these choices.

Question 29

If a simple random sample of 300 observations is taken from a population whose proportion p = 0.6, then the expected value of the sample proportion is 0.60.

Question 30

The standard error of the sampling distribution of the sample proportion , when the sample size n = 100 and the population proportion p = 0.30, is 0.0021.

Question 31

The standard deviation of the sampling distribution of is also called the:

A) central limit theorem.
B) population standard deviation.
C) finite population correction factor.
D) standard error of the sample mean.

Question 32

In an effort to identify the true proportion of college freshman who are under 18 years of age, a random sample of 500 freshmen was taken. Fifty of them were under the age of 18. The value 0.10 is a point estimate of the true proportion of freshman under age 18.

Question 33

The finite population correction factor should be used:

A) whenever we are sampling from an infinite population.
B) whenever we are sampling from a finite population.
C) whenever the sample size is large compared to the population size.
D) whenever the sample size is small compared to the population size.

Question 34

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:

A) 45 and 9
B) 45/81 and 9/81
C) 45 and 9/ $\surd$ 81
D) 45/ $\surd$ 81 and 9/ $\surd$ 81

Question 35

In general, the binomial probability P(X $\ge$ x) is approximated by the area under the normal curve to the left of x - .5.

Question 36

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:

A) 25
B) 25/64
C) 25/ $\surd$ 64
D) None of these choices.

Question 37

A sample of size 40 is taken from an infinite population whose mean and standard deviation are 68 and 12, respectively. The probability that the sample mean is larger than 70 equals

A) P(Z > 70)
B) P(Z > 2)
C) P(Z > 0.17)
D) P(Z > 1.05)

Question 38

If all possible samples of size n are drawn from an infinite population with a mean of 15 and a standard deviation of 5, then the standard error of the sample mean equals 1.0 for samples of size:

A) 5
B) 15
C) 25
D) None of these choices.

Question 39

In general, the binomial probability P(X $\le$ x) is approximated by the area under the normal curve to the left of x + .5.

Question 40

An infinite population has a mean of 40 and a standard deviation of 15. A sample of size 100 is taken at random from this population. The standard error of the sample mean equals:

A) 15
B) 15/ $\surd$ 100
C) 15/100
D) None of these choices.

Question 41

Given that X is a binomial random variable with very large n, the binomial probability P(X $\ge$ 5) is approximated by the area under a normal curve to the right of

A) 4.5
B) 5.5
C) 4
D) 6

Question 42

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean is called the:

A) standard error of.
B) expected value of.
C) sampling distribution of.
D) normal distribution.

Question 43

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true?

A) The population standard deviation decreases.
B) The standard error of the sample mean decreases.
C) The population standard deviation increases.
D) The standard error of the sample mean increases.

Question 44

Sampling distributions describe the distributions of:

A) sample statistics.
B) population parameters.
C) both parameters and statistics
D) None of these choices.

Question 45

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if:

A) the sample size n is greater than 30.
B) the population proportion p is close to 0.50.
C) the underlying population is normal.
D) np and n(1-p) are both greater than or equal to 5.

Question 46

The standard error of the mean:

A) is never larger than the standard deviation of the population.
B) decreases as the sample size increases.
C) measures the variability of the mean from sample to sample.
D) All of these choices are true.

Question 47

For a sample size of 1, the sampling distribution of the mean is normally distributed:

A) regardless of the shape of the population.
B) only if the population values are larger than 30.
C) only if the population is normally distributed.
D) None of these choices.

Question 48

Suppose X has a distribution that is not normal. The Central Limit Theorem is important in this case because:

A) it says the sampling distribution ofis approximately normal for any sample size.
B) it says the sampling distribution ofis approximately normal if n is large enough.
C) it says the sampling distribution ofis exactly normal, for any sample size.
D) None of these choices.

Question 49

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean will be:

A) normal for all values of n.
B) normal only for n > 30.
C) approximately normal for all values of n.
D) approximately normal only for n > 30.

Question 50

The owner of a meat market has an assistant who has determined that the weights of roasts are normally distributed, with a mean of 3.2 pounds and standard deviation of 0.8 pounds. If a sample of 25 roasts yields a mean of 3.6 pounds, what is the Z-score for this sample mean?

A) -2.50
B) 2.50
C)-0.50
D) None of these choices.

Question 51

The standard error of the mean for a sample of 100 is 25. In order to cut the standard error of the mean in half (to 12.5) we must:

A) increase the sample size to 200.
B) decrease the sample size to 50.
C) keep the sample size at 100 and change something else.
D) None of these choices.

Question 52

For sample sizes greater than 30, the sampling distribution of the mean is approximately normally distributed:

A) regardless of the shape of the population.
B) only if the shape of the population is symmetric.
C) only if the population is normally distributed.
D) None of these choices.

Question 53

Which of the following statements about the sampling distribution of is NOT true?

A) It is generated by taking all possible samples of size n and computing their sample means.
B) Its mean is equal to the population mean $\mu$ .
C) Its standard deviation is equal to the population standard deviation $\sigma$ .
D) All of these choices are true.

Question 54

Which of the following is true regarding the sampling distribution of the mean for a large sample size? Assume the population distribution is not normal.

A) It has the same shape, mean and standard deviation as the population.
B) It has the same mean as the population, but a different shape and standard deviation.
C) It the same mean and standard deviation as the population, but a different shape.
D) It has the same shape and mean as the population, but a different standard deviation.

Question 55

The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed, with a mean of 3.2 pounds and standard deviation of 0.84 pounds. If a sample of 16 fish is taken, what is the standard error of the mean weight?

A) 0.840
B) 0.053
C) 0.210
D) None of these choices.

Question 56

Suppose that 100 items are drawn from a population of manufactured products and the weight, X, of each item is recorded. Prior experience has shown that the weight has a non-normal probability distribution with $\mu$ = 8 ounces and $\sigma$ = 3 ounces. Which of the following is true about the sampling of ?

A) Its mean is 8 ounces.
B) Its standard error is 0.3 ounces.
C) Its shape is approximately normal.
D) All of these choices are true.

Question 57

Given that X is a binomial random variable with very large n, the binomial probability P(X = 5) is approximated by the area under a normal curve between

A) 5 and 5
B) 4 and 6
C) 4.5 and 5.5
D) None of these choices.

Question 58

The expected value of the sampling distribution of the sample mean equals the population mean $\mu$ :

A) only when the population is normally distributed.
B) only when the sample size is large.
C) only when the population is infinite.
D) for all populations.

Question 59

Given that X is a binomial random variable with very large n, the binomial probability P(X $\le$ 5) is approximated by the area under a normal curve to the left of

A) 5
B) -5
C) 5.5
D) 4.5

Question 60

Which of the following is true about the sampling distribution of the sample mean?

A) Its mean is always equal to
B) Its standard error is always equal to the population standard deviation $\sigma$ .
C) Its shape is exactly normal if n is large enough.
D) None of these choices.

Question 61

The standard deviation of is also called the:

A) standard error of the sample proportion.
B) standard deviation of the population.
C) standard deviation of the binomial.
D) None of these choices.

Question 62

As the number of throws of a fair die increases, the probability that the sample mean is close to (the number) ____________________ increases.

Question 63

A sample of size 200 is taken at random from an infinite population. Given that the population proportion is 0.60, the probability that the sample proportion is between 0.58 and 0.62 is:

A) 0.4314
B) 0.0320
C) 0.0200
D) None of these choices.

Question 64

A sample of size 200 is taken at random from an infinite population. Given that the population proportion is 0.60, the probability that the sample proportion is less than 0.58 is

A) 0.281
B) 0.719
C) 0.580
D) 0.762

Question 65

The width of the sampling distribution of gets ____________________ as the sample size increases.

Question 66

The standard error of the sample proportion gets larger as:

A) p approaches 0
B) p approaches 0.50
C) p approaches 1.00
D) None of these choices.

Question 67

Suppose that the probability p of success on any trail of a binomial distribution equals 0.90. Then for which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

A) 35
B) 45
C) 55
D) All of these choices are true.

Question 68

Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequate approximation of the binomial distribution if

A) np $\le$ 5 and n(1 - p) $\le$ 5
B) np $\ge$ 5 and np(1 -p) $\ge$ 5
C) np $\ge$ 5 and n(1 - p) $\ge$ 5
D) None of these choices.

Question 69

As n gets larger, the sampling distribution of becomes increasingly bell shaped. This phenomenon is due to the ____________________.

Question 70

The accuracy of the approximation of with a normal distribution depends on the probability distribution of the ____________________ and on the sample ____________________.

Question 71

If the population is normal, then is normally distributed for __________________ values of n.

Question 72

A sample of size 200 is taken at random from an infinite population. Given that the population proportion is 0.60, the probability that the sample proportion is greater than 0.58 is:

A) 0.281
B) 0.719
C) 0.580
D) 0.762

Question 73

The estimator of the probability of success in a binomial distribution is: ____________________.

Question 74

Under certain conditions where n is large enough, you can approximate the ____________________ distribution using the ____________________ distribution.

Question 75

The finite population correction is not needed if the population size is large relative to the sample size. As a rule of thumb, we will treat any population that is at least ____________________ times larger than the sample size as large.

Question 76

A sample of 250 observations is selected at random from an infinite population. Given that the population proportion is .25, the standard error of the sampling distribution of the sample proportion is:

A) 0.0274
B) 0.5000
C) 0.0316
D) 0.0548

Question 77

A randomly selected value of is likely to be ____________________ to(than) the mean of the population than is a randomly selected value of X.

Question 78

The variance of the sampling distribution of is ____________________ the variance of the population we're sampling from, for any n > 1.

Question 79

Because the value of the ____________________ varies from sample to sample, we can regard it as a new random variable, created by sampling.

Question 80

As n gets ____________________, the shape of the sampling distribution of becomes increasingly bell shaped.