Deck 9: Sampling Distributions

Question 1

If two random samples of sizes n₁ and n₂ are selected independently from two non-normally distributed populations, then the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is

A)always non-normal
B)always normal
C)approximately normal only if n₁ and n₂ are both larger than or equal to 30
D)approximately normal regardless of n₁ and n₂

Answer 1

approximately normal only if n₁ and n₂ are both larger than or equal to 30

Question 2

NARRBEGIN: CFO's Salary
CFO's Salary
Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected.NARREND

-Two samples are selected independently from two normal populations and the mean and standard error of the sampling distribution of $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ are 32 and 38.72, respectively. Calculate P( $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ > 0).

Answer 2

P( $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ > 0) = P (Z > -0.83) = 0.7967

Question 3

The expected value of the sampling distribution of $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is $\mu _ { \bar { g } _ { 1 } } - \bar { x } _ { 2 } = \mu _ { 1 } - \mu _ { 2 }$ where $\mu$ _i is the mean of population i (i = 1, 2).

Answer 3

True

Question 4

NARRBEGIN: CFO's Salary
CFO's Salary
Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected.NARREND
{CFO's Salary Narrative} What is the probability that the sample mean salary of male CFO's will not exceed that of the female CFO's?

Question 5

If two random samples of sizes 30 and 32 are selected independently from two populations with means 121 and 109, then the mean of the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , equals 12.

Question 6

The mean of the difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is equal to the mean of the difference $\bar { X } _ { 2 } - \bar { X } _ { 1 }$ .

Question 7

If two random samples of sizes 30 and 36 are selected independently from two populations with means 78 and 85, and standard deviations 12 and 15, respectively, then the mean of the difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is equal to:

A)-7
B)7
C)(78 - 85) / (30 -36) = 1.17
D)78/30 - 85/36 = 0.24

Question 8

NARRBEGIN: CFO's Salary
CFO's Salary
Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected.NARREND
{CFO's Salary Narrative} Find the expected value of the sample mean difference.

Question 9

If two random samples of sizes n₁ and n₂ are selected independently from two populations with means $\mu$ ₁ and $\mu$ ₂, then the mean of $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ equals:

A) $\mu$ ₁ + $\mu$ ₂
B) $\mu$ ₁ - $\mu$ ₂
C) $\mu$ ₁ / $\mu$ ₂
D) $\mu$ ₁ $\mu$ ₂

Question 10

If two samples are selected independently from two non-normal populations, then the sampling distribution of $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is only approximately normal provided that either n₁ or n₂ is 30 or more.

Question 11

If two random samples of sizes 30 and 36 are selected independently from two populations with means 78 and 85, and standard deviations 12 and 15, respectively, then the standard error of the difference $\bar { X } _ { 1 }$ and $\bar { X } _ { 2 }$ is equal to:

A)0.904
B)3.324
C)3.391
D)0.833

Question 12

NARRBEGIN: CFO's Salary
CFO's Salary
Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected.NARREND
Two random samples of sizes 30 and 36 are selected independently from two populations with means 80 and 88, and standard deviations 15 and 20, respectively.
a.Find the standard error of the difference betweenand.
b.Find the probability that the mean of the first sample is smaller than the mean of the second sample.

Question 13

NARRBEGIN: CFO's Salary
CFO's Salary
Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected.NARREND
{CFO's Salary Narrative} What is the sampling distribution of the sample mean difference ? Explain.

Question 14

The standard deviation of $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is also called the:

A)standard error of the difference between two sample means.
B)standard deviation of the difference between the population means.
C)normal approximation to the difference of two binomial random variables.
D)None of these choices.

Question 15

The standard error of the difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is equal to the standard error of the difference $\bar { X } _ { 2 } - \bar { X } _ { 1 }$ .

Question 16

If two random samples of size 36 each are selected independently from two populations with variances 25 and 16, then the standard error of the sampling distribution of the sample mean difference, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is 5-4 = 1.

Question 17

The standard error of the difference between sample means, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is calculated by the formula $\sigma _ { \bar { x } _ { 1 } - \bar { x } _ { 2 } } = \sqrt { \sigma _ { 1 } ^ { 2 } / n _ { 1 } - \sigma _ { 2 } ^ { 2 } / n _ { 2 } }$ where $\sigma _ { i } ^ { 2 }$ is the variance of population i (i = 1, 2).

Question 18

If two populations are normally distributed, the sampling distribution of the difference in the sample means, $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ , is:

A)approximately normal for any sample sizes.
B)approximately normal if both sample sizes are large.
C)exactly normal only if both sample sizes are large.
D)exactly normal for any sample sizes.

Question 19

NARRBEGIN: Temporary Worker Producti
Temporary Worker Productivity
A temporary worker productivity is normally distributed. One worker produces an average of 84 units per day with a standard deviation of 24. Another worker produces at an average rate of 74 per day with a standard deviation of 25.NARREND
{Temporary Worker Productivity Narrative} What is the probability that in any single day worker 1 will outproduce worker 2?

Question 20

NARRBEGIN: CFO's Salary
CFO's Salary
Suppose that the starting salaries of female CFO's have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO's are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO's and a random sample of 40 male CFO's are selected.NARREND
{CFO's Salary Narrative} Find the standard error of the sample mean difference.

Question 21

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 22

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 23

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 $\tilde{ p }$ is 0.60.

Question 24

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 25

NARRBEGIN: Temporary Worker Producti
Temporary Worker Productivity
A temporary worker productivity is normally distributed. One worker produces an average of 84 units per day with a standard deviation of 24. Another worker produces at an average rate of 74 per day with a standard deviation of 25.NARREND
{Temporary Worker Productivity Narrative} What is the probability that during one week (5 working days), worker 1 will outproduce worker 2 on average?

Question 26

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 27

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 28

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 29

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

Question 30

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

Question 31

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 32

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 33

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

Question 34

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

Question 35

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 36

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 37

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 38

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 39

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 40

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 41

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

Question 42

The continuity correction factor adds and subtracts the number ____________________ to/from x to find P(X = x).

Question 43

The normal approximation to the binomial distribution gets better and better as ____________________ increases.

Question 44

NARRBEGIN: DVD Rental Store
DVD Rental Store
A DVD rental store wants to know what proportion of its customers are under age 21. A simple random sample of 500 customers was taken, and 375 of them were under age 21. Presume that the true population proportion of customers under age 21 is 0.68.NARREND

-{DVD Rental Store Narrative} Describe the shape of the sampling distribution of proportion of customers who are under age 21.

Question 45

Suppose it is known that 60% of radio listeners at a particular college are smokers. A sample of 500 students from the college is selected at random. Approximate the probability that at least 280 of these students are radio listeners.

Question 46

NARRBEGIN: Barack Obama
Barack Obama
Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Barack Obama in the year 2012 presidential elections is 0.65. A sample of 500 voters is selected at random.NARREND

-Let X be a binomial random variable with n = 100 and p = 0.7. Approximate the following probabilities, using the normal distribution.
a.P(X = 75)
b.P(X $\le$ 70)
c.P(X > 70)

Question 47

What is the probability that you get more than 200 heads if you flip a coin 400 times?

Question 48

The variance of is ____________________.

Question 49

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 50

NARRBEGIN: DVD Rental Store
DVD Rental Store
A DVD rental store wants to know what proportion of its customers are under age 21. A simple random sample of 500 customers was taken, and 375 of them were under age 21. Presume that the true population proportion of customers under age 21 is 0.68.NARREND
{DVD Rental Store Narrative} Find the mean and standard deviation of .

Question 51

The standard deviation of $\tilde{ p }$ 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 52

NARRBEGIN: Barack Obama
Barack Obama
Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Barack Obama in the year 2012 presidential elections is 0.65. A sample of 500 voters is selected at random.NARREND
{Barack Obama Narrative} Find the expected value and the standard deviation of the sample proportion .

Question 53

NARRBEGIN: DVD Rental Store
DVD Rental Store
A DVD rental store wants to know what proportion of its customers are under age 21. A simple random sample of 500 customers was taken, and 375 of them were under age 21. Presume that the true population proportion of customers under age 21 is 0.68.NARREND
{DVD Rental Store Narrative} What is the probability that the sample proportion is within 0.03 of the true proportion of customers who are under age 21?

Question 54

NARRBEGIN: Barack Obama
Barack Obama
Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Barack Obama in the year 2012 presidential elections is 0.65. A sample of 500 voters is selected at random.NARREND

-{Barack Obama Narrative} Assume that Gore is correct and p = 0.65. What is the sampling distribution of the sample proportion $\tilde{ p }$ ? Explain.

Question 55

To find the binomial probability P(X $\le$ 4) we calculate the area under the normal curve to the ____________________ (left/right) of the number ____________________.

Question 56

NARRBEGIN: Barack Obama
Barack Obama
Bill Clinton, the former President of the United States, believes that the proportion of voters who will vote for Barack Obama in the year 2012 presidential elections is 0.65. A sample of 500 voters is selected at random.NARREND
{Barack Obama Narrative} What is the probability that the number of voters in the sample who will vote for Barack Obama in the year 2012 is between 340 and 350?

Question 57

The expected value of is ____________________.

Question 58

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

Question 59

The probability of success on any trial of a binomial experiment is 20%. Find the probability that the proportion of success in a sample of 400 is:
a.less than 18%.
b.more than 18%.
c.between 18% and 22%.

Question 60

The standard deviation of is also called the ____________________ of the sample proportion.

Question 61

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 62

NARRBEGIN: DVD Rental Store
DVD Rental Store
A DVD rental store wants to know what proportion of its customers are under age 21. A simple random sample of 500 customers was taken, and 375 of them were under age 21. Presume that the true population proportion of customers under age 21 is 0.68.NARREND

-Given a binomial random variable with n = 15 and p = .40, find the exact probabilities of the following events and their normal approximations.
a.X = 6
b.X $\ge$ 9
c.X $\le$ 10

Question 63

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 64

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

Question 65

NARRBEGIN: Graduate Internships
Graduate Internships
The chairman of the Biology department in a certain college believes that 70% of the department's graduate internships are given to international students. A random sample of 50 graduate interns is taken.NARREND
{Graduate Internships Narrative} Find the expected value and the standard error of the sampling distribution of .

Question 66

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 67

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

Question 68

If all possible samples of size n are drawn from a normal population, the probability distribution of the sample mean $\bar { X }$ is an exact normal distribution.

Question 69

NARRBEGIN: Graduate Internships
Graduate Internships
The chairman of the Biology department in a certain college believes that 70% of the department's graduate internships are given to international students. A random sample of 50 graduate interns is taken.NARREND

-{Graduate Internships Narrative} What is the probability that the sample proportion $\tilde{ p }$ is within $\pm$ .05 of the population proportion p?

Question 70

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 71

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 72

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.

Question 73

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

Question 74

A sample of size 25 is selected from a population of size 75. The finite population correction needed to find the standard error of $\bar { X }$ .

Question 75

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 $\bar { X }$ .

Question 76

NARRBEGIN: Graduate Internships
Graduate Internships
The chairman of the Biology department in a certain college believes that 70% of the department's graduate internships are given to international students. A random sample of 50 graduate interns is taken.NARREND
{Graduate Internships Narrative} What is the probability that the sample proportion is between 0.65 and 0.73?

Question 77

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.

Question 78

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 79

NARRBEGIN: Graduate Internships
Graduate Internships
The chairman of the Biology department in a certain college believes that 70% of the department's graduate internships are given to international students. A random sample of 50 graduate interns is taken.NARREND

-{Graduate Internships Narrative} Assume that the chairman is correct and p = 0.70. What is the sampling distribution of the sample proportion $\tilde{ p }$ ? Explain.

Question 80

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.