Deck 8: Sampling Distributions and Estimation

Question 1

The Central Limit Theorem says that a histogram of the sample means will have a bell shape, even if the population is skewed and the sample is small.

Answer 1

False

Question 2

A 90 percent confidence interval will be wider than a 95 percent confidence interval, ceteris paribus.

Answer 2

False

Question 3

In constructing confidence intervals, it is conservative to use the z distribution when n ≥ 30.

Answer 3

False

Question 4

The Central Limit Theorem guarantees an approximately normal sampling distribution when n is sufficiently large.

Question 5

In constructing a confidence interval for a mean, the width of the interval is dependent on the sample size, the confidence level, and the population standard deviation.

Question 6

The expected value of an unbiased estimator is equal to the parameter whose value is being estimated.

Question 7

The efficiency of an estimator depends on the variance of the estimator's sampling distribution.

Question 8

All estimators are biased since sampling error always exists to some extent.

Question 9

The confidence level refers to the procedure used to construct the confidence interval, rather than to the particular confidence interval we have constructed.

Question 10

When the sample standard deviation is used to construct a confidence interval for the mean, we would use the Student's t distribution instead of the normal distribution.

Question 11

As n increases, the standard error decreases.

Question 12

As long as the sample is more than one item, the standard error of the sample mean will be smaller than the standard deviation of the population.

Question 13

An estimator must be unbiased if you are to use it for statistical analysis.

Question 14

A sample of size 5 shows a mean of 45.2 and a sample standard deviation of 6.4. The standard error of the sample mean is approximately 2.86.

Question 15

The Central Limit Theorem says that, if n exceeds 30, the population will be normal.

Question 16

For a sample size of 20, a 95 percent confidence interval using the t distribution would be wider than one constructed using the z distribution.

Question 17

In constructing a confidence interval for the mean, the z distribution provides a result nearly identical to the t distribution when n is large.

Question 18

In comparing estimators, the more efficient estimator will have a smaller standard error.

Question 19

As n increases, the width of the confidence interval will decrease, ceteris paribus.

Question 20

A higher confidence level leads to a narrower confidence interval, ceteris paribus.

Question 21

The sample proportion is always the midpoint of a confidence interval for the population proportion.

Question 22

The distribution of the sample proportion p = x/n is normal when n ≥ 30.

Question 23

Based on the Rule of Three, if no events occur in n independent trials we can set the upper 95 percent confidence bound at 3/n.

Question 24

To estimate a proportion with a 4 percent margin of error and a 95 percent confidence level, the required sample size is over 800.

Question 25

The standard deviation of the sample proportion p = x/n increases as n increases.

Question 26

The standard error of the sample proportion does not depend on the confidence level.

Question 27

In a sample size calculation, if the confidence level decreases, the size of the sample needed will increase.

Question 28

The sample standard deviation s is halfway between the lower and upper confidence limits for the population σ (i.e., the confidence interval is symmetric around s).

Question 29

The Central Limit Theorem can be applied to the sample proportion.

Question 30

The statistic p = x/n may be assumed normally distributed when np ≥ 10 and n(1 - p) ≥ 10.

Question 31

To estimate π, you typically need a sample size equal to at least 5 percent of your population.

Question 32

The standard error of the sample proportion is largest when π = .50.

Question 33

A 95 percent confidence interval constructed around p will be wider than a 90 percent confidence interval.

Question 34

To calculate the sample size needed for a survey to estimate a proportion, the population standard deviation σ must be known.

Question 35

To narrow the confidence interval for π, we can either increase n or decrease the level of confidence.

Question 36

The Student's t distribution is always symmetric and bell-shaped, but its tails lie above the normal.

Question 37

Assuming that π = .50 is a quick and conservative approach to use in a sample size calculation for a proportion.

Question 38

Ceteris paribus, the narrowest confidence interval for π is achieved when p = .50.

Question 39

To estimate the required sample size for a proportion, one method is to take a small pilot sample to estimate π and then apply the sample size formula.

Question 40

The confidence interval half-width when π = .50 is called the margin of error.

Question 41

In constructing a confidence interval, the finite population correction factor (FPCF) can be ignored if samples of 12 items are drawn from a population of 300 items.

Question 42

Sampling variation is not controllable by the statistician.

Question 43

To determine a 72 percent level of confidence for a proportion, the value of z is approximately:

A)± 1.65
B)± 0.77
C)± 1.08
D)± 1.55

Question 44

The width of a confidence interval for μ is not affected by:

A)the sample size.
B)the confidence level.
C)the standard deviation.
D)the sample mean.

Question 45

In constructing a confidence interval for a mean with unknown variance with a sample of 25 items, Bob used z instead of t. "Well, at least my interval will be wider than necessary, so it was a conservative error," said he. Is Bob's statement correct?

A)Yes.
B)No.
C)It depends on μ.

Question 46

We could narrow a 95 percent confidence interval by:

A)using 99 percent confidence.
B)using a larger sample.
C)raising the standard error.

Question 47

A 99 percent confidence interval has higher confidence but less precision than a 95 percent confidence interval.

Question 48

The finite population correction factor (FPCF) can be ignored if n = 7 and N = 700.

Question 49

Approximately 95 percent of the population X values will lie within the 95 percent confidence interval for the mean.

Question 50

A random sample of 16 ATM transactions at the Last National Bank of Flat Rock revealed a mean transaction time of 2.8 minutes with a standard deviation of 1.2 minutes. The width (in minutes) of the 95 percent confidence interval for the true mean transaction time is:

A)± 0.639
B)± 0.588
C)± 0.300
D)± 2.131

Question 51

The finite population correction factor (FPCF) can be ignored when the sample size is large relative to the population size.

Question 52

To estimate the average annual expenses of students on books and class materials, a sample of size 36 is taken. The sample mean is $850 and the sample standard deviation is $54. A 99 percent confidence interval for the population mean is:

A)$823.72 to $876.28
B)$832.36 to $867.64
C)$826.82 to $873.18
D)$825.48 to $874.52

Question 53

A highway inspector needs an estimate of the mean weight of trucks crossing a bridge on the interstate highway system. She selects a random sample of 49 trucks and finds a mean of 15.8 tons with a sample standard deviation of 3.85 tons. The 90 percent confidence interval for the population mean is:

A)14.72 to 16.88 tons.
B)14.90 to 16.70 tons.
C)14.69 to 16.91 tons.
D)14.88 to 16.72 tons.

Question 54

A sampling distribution describes the distribution of:

A)a parameter.
B)a statistic.
C)either a parameter or a statistic.
D)neither a parameter nor a statistic.

Question 55

The Central Limit Theorem (CLT) implies that:

A)the population will be approximately normal if n ≥ 30.
B)repeated samples must be taken to obtain normality.
C)the distribution of the mean is approximately normal for large n.
D)the mean follows the same distribution as the population.

Question 56

The sample mean is not a random variable when the population parameters are known.

Question 57

The owner of Torpid Oaks B&B wanted to know the average distance its guests had traveled. A random sample of 16 guests showed a mean distance of 85 miles with a standard deviation of 32 miles. The 90 percent confidence interval (in miles) for the mean is approximately:

A)(71.0, 99.0)
B)(71.8, 98.2)
C)(74.3, 95.7)
D)(68.7, 103.2)

Question 58

As the sample size increases, the standard error of the mean:

A)increases.
B)decreases.
C)may increase or decrease.

Question 59

Which statement is most nearly correct, other things being equal?

A)Doubling the sample size will cut the standard error of the mean in half.
B)The standard error of the mean depends on the population size.
C)Quadrupling the sample size roughly halves the standard error of the mean.
D)The standard error of the mean depends on the confidence level.

Question 60

The owner of Limp Pines Resort wanted to know the average age of its clients. A random sample of 25 tourists is taken. It shows a mean age of 46 years with a standard deviation of 5 years. The width of a 98 percent CI for the true mean client age is approximately:

A)± 1.711 years.
B)± 2.326 years.
C)± 2.492 years.
D)± 2.797 years.

Question 61

An airport traffic analyst wants to estimate the proportion of daily takeoffs by small business jets (as opposed to commercial passenger jets or other aircraft) with an error of ± 4 percent with 90 percent confidence. What sample size should the analyst use?

A)385
B)601
C)410
D)423

Question 62

A company wants to estimate the time its trucks take to drive from city A to cityB. The standard deviation is known to be 12 minutes. What sample size is required in order that the error will not exceed ±2 minutes, with 95 percent confidence?

A)12 observations
B)139 observations
C)36 observations
D)129 observations

Question 63

A random sample of 160 commercial customers of PayMor Lumber revealed that 32 had paid their accounts within a month of billing. Can normality be assumed for the sample proportion?

A)Yes.
B)No.
C)Need more information to say.

Question 64

Which statement is incorrect? Explain.

A)If p = .50 and n = 100, the standard error of the sample proportion is .05.
B)In a sample size calculation for estimating π, it is conservative to assume π = .50.
C)If n = 250 and p = .06, we cannot assume normality in a confidence interval for π.

Question 65

Ersatz Beneficial Insurance wants to estimate the cost of damage to cars due to accidents. The standard deviation of the cost is known to be $200. They want to estimate the mean cost using a 95 percent confidence interval within ± $10. What is the minimum sample size n?

A)1083
B)4002
C)1537
D)2301

Question 66

A poll showed that 48 out of 120 randomly chosen graduates of California medical schools last year intended to specialize in family practice. What is the width of a 90 percent confidence interval for the proportion that plan to specialize in family practice?

A)± .0447
B)± .0736
C)± .0876
D)± .0894

Question 67

What is the approximate width of a 90 percent confidence interval for the true population proportion if there are 12 successes in a sample of 25?

A)± .196
B)± .164
C)± .480
D)± .206

Question 68

In a large lecture class, the professor announced that the scores on a recent exam were normally distributed with a range from 51 to 87. Using the Empirical Rule μ ± 3σ to estimate σ, how many students would you need to sample to estimate the true mean score for the class with 90 percent confidence and an error of ±2?

A)About 17 students
B)About 35 students
C)About 188 students
D)About 25 students

Question 69

In a random sample of 810 women employees, it is found that 81 would prefer working for a female boss. The width of the 95 percent confidence interval for the proportion of women who prefer a female boss is:

A)± .0288
B)± .0105
C)± .0207
D)± .0196

Question 70

A university wants to estimate the average distance that commuter students travel to get to class with an error of ± 3 miles and 90 percent confidence. What sample size would be needed, assuming that travel distances are normally distributed with a range of X = 0 to X = 50 miles, using the Empirical Rule μ ± 3σ to estimate σ.

A)About 28 students
B)About 47 students
C)About 30 students
D)About 21 students

Question 71

A random sample of 160 commercial customers of PayMor Lumber revealed that 32 had paid their accounts within a month of billing. The 95 percent confidence interval for the true proportion of customers who pay within a month would be:

A)0.148 to 0.252
B)0.138 to 0.262
C)0.144 to 0.256
D)0.153 to 0.247

Question 72

A marketing firm is asked to estimate the percentage of existing customers who would purchase a "digital upgrade" to their basic cable TV service. The firm wants 99 percent confidence and an error of ± 5 percent. What is the required sample size (to the next higher integer)?

A)664
B)625
C)801
D)957

Question 73

Which of the following is not a characteristic of the t distribution?

A)It is a continuous distribution.
B)It has a mean of 0.
C)It is a symmetric distribution.
D)It approaches z as degrees of freedom decrease.

Question 74

Jolly Blue Giant Health Insurance (JBGHI) is concerned about rising lab test costs and would like to know what proportion of the positive lab tests for prostate cancer are actually proven correct through subsequent biopsy. JBGHI demands a sample large enough to ensure an error of ± 2 percent with 90 percent confidence. What is the necessary sample size?

A)4,148
B)2,401
C)1,692
D)1,604

Question 75

Professor York randomly surveyed 240 students at Oxnard University and found that 150 of the students surveyed watch more than 10 hours of television weekly. Develop a 95 percent confidence interval to estimate the true proportion of students who watch more than 10 hours of television each week. The confidence interval is:

A).533 to .717
B).564 to .686
C).552 to .698
D).551 to .739

Question 76

A financial institution wishes to estimate the mean balances owed by its credit card customers. The population standard deviation is $300. If a 99 percent confidence interval is used and an interval of ± $75 is desired, how many cardholders should be sampled?

A)3382
B)629
C)87
D)107

Question 77

Using the conventional polling definition, find the margin of error for a customer satisfaction survey of 225 customers who have recently dined at Applebee's.

A)± 5.0 percent
B)± 4.2 percent
C)± 7.1 percent
D)± 6.5 percent

Question 78

Last week, 108 cars received parking violations in the main university parking lot. Of these, 27 had unpaid parking tickets from a previous violation. Assuming that last week was a random sample of all parking violators, find the 95 percent confidence interval for the percentage of parking violators that have prior unpaid parking tickets.

A)18.1 to 31.9 percent.
B)16.8 to 33.2 percent.
C)15.3 to 34.7 percent.
D)19.5 to 30.5 percent.

Question 79

What is the approximate width of an 80 percent confidence interval for the true population proportion if there are 12 successes in a sample of 80?

A)± .078
B)± .066
C)± .051
D)± .094

Question 80

In constructing a 95 percent confidence interval, if you increase n to 4n, the width of your confidence interval will be (assuming other things remain the same):

A)about 25 percent of its former width.
B)about two times wider.
C)about 50 percent of its former width.
D)about four times wider.