Deck 11: Introduction to Hypothesis Testing

Question 1

As the alternative value of $\mu$ increases, so does the power of the test.

Answer 1

True

Question 2

For a given level of significance, if the sample size increases, the probability of a Type II error will:

A)remain the same.
B)increase.
C)decrease.
D)be equal to 1.0 regardless of $\alpha$ .

Answer 2

decrease.

Question 3

If a sample size is increased at a given $\alpha$ level, the probability of committing a Type II error is increased.

Answer 3

False

Question 4

For a given level of significance, if the sample size is increased, the power of the test will increase.

Question 5

For a given level of significance, if the sample size is increased, the probability of committing a Type II error will ____________________.

Question 6

For a given level of significance, if the sample size is increased, the power of the test will ____________________.

Question 7

The operating characteristic curve plots the values of $\beta$ (the probability of committing a Type II error) versus the values of the population mean $\mu$ .

Question 8

One way of expressing how well a test performs is to report its power--the probability of detecting a false null hypothesis.

Question 9

The power of the test refers to the probability of rejecting a false null hypothesis.

Question 10

For a given sample size n, if the level of significance $\alpha$ is decreased, the power of the test will:

A)increase.
B)decrease.
C)remain the same.
D)Not enough information to tell.

Question 11

If the probability of committing a Type I error for a given test is decreased, then for a fixed sample size n, the probability of committing a Type II error will:

A)decrease.
B)increase.
C)stay the same.
D)Not enough information to tell.

Question 12

Which of the following statements is false regarding the operating characteristic (OC) curve?

A)The OC curve plots the values of $\beta$ versus the values of $\mu$ .
B)The OC curve plots the values of $\alpha$ versus the values of $\beta$ .
C)The OC curve can be useful in selecting a sample size n.
D)None of these choices.

Question 13

For a given level of significance $\alpha$ , if the sample size n is increased, the probability of a Type II error $\beta$ will:

A)decrease.
B)increase.
C)remain the same.
D)Not enough information to tell.

Question 14

For a given sample size, the probability of committing a Type II error will increase when the probability of committing a Type I error is reduced.

Question 15

The power of a test is the probability that a true null hypothesis will be rejected.

Question 16

If we want to compute the probability of a Type II error, which of the following statements is false?

A)We need to know the significance level $\alpha$ .
B)We need to know the sample size n.
C)We need to know the alternative value of the population mean $\mu$ .
D)All of these choices are true.

Question 17

The power of a test is denoted by:

A) $\alpha$
B) $\beta$
C)1 - $\alpha$
D)1 - $\beta$

Question 18

There is a direct relationship between the power of a test and the probability of a Type II error.

Question 19

The power of a test is measured by its capability of:

A)rejecting a null hypothesis that is true.
B)not rejecting a null hypothesis that is true.
C)rejecting a null hypothesis that is false.
D)not rejecting a null hypothesis that is false.

Question 20

For a given level of significance, if the sample size is increased, the probability of committing a Type II error will increase.

Question 21

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Calculate $\beta$ , the probability of a Type II error when $\mu$ = 1000 and $\alpha$ = 0.10.

Question 22

To calculate the probability of a(n) ____________________ error you need to specify a value of $\mu$ other than the one given in the null hypothesis.

Question 23

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Recalculate $\beta$ if n is increased from 25 to 40.

Question 24

As the sample size increases, the operating characteristic curves drop down to zero at a(n) ____________________ rate.

Question 25

To test the hypotheses: H₀: $\mu$ = 40 vs. H₁: $\mu$ $\neq$ 40, we draw a random sample of size 16 from a normal population whose standard deviation is 5. If we set $\alpha$ = 0.01, find the probability of committing a Type II error when $\mu$ = 37.

Question 26

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Calculate the power of the test when $\mu$ = 1000 and $\alpha$ = 0.10.

Question 27

Calculate the probability of a Type II error for the hypothesis test: H₀: $\mu$ = 50 vs. H₁: $\mu$ > 50, given that $\mu$ = 55, $\alpha$ = 0.05, $\sigma$ = 10, and n = 16.

Question 28

To increase the power of a test, ____________________ the sample size.

Question 29

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Recalculate $\beta$ if $\alpha$ is lowered from 0.10 to 0.05.

Question 30

By ____________________ the significance level, you increase the probability of a Type II error.

Question 31

The ____________________ of a test is a measure of its performance.

Question 32

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Interpret the meaning of the power of the test.

Question 33

A p-value is usually set at 0.05.

Question 34

A(n) ____________________ characteristic curve plots the probability of a Type II error for various alternative values of ____________________.

Question 35

During the last energy crisis, a government official claimed that the average car owner refills the tank when there is more than 3 gallons left. To check the claim, 10 cars were surveyed as they entered a gas station. The amount of gas remaining before refill was measured and recorded as follows (in gallons): 3, 5, 3, 2, 3, 3, 2, 6, 4, and 1. Assume that the amount of gas remaining in tanks is normally distributed with a standard deviation of 1 gallon. Compute the probability of a Type II error and the power of the test if the true average amount of gas remaining in tanks is 3.5 gallons.

Question 36

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Review the results of the previous questions. What is the effect of decreasing the significance level on the value on $\beta$ ?

Question 37

The p-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true.

Question 38

Probabilities for Type I and Type II errors are actually ____________________ probabilities.

Question 39

The power of a test plus the probability of a Type II error equals ____________________.

Question 40

NARRBEGIN: Rechargeable Batteries
Rechargeable Batteries
A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours). In testing the hypotheses, H₀: $\mu$ = 950 hours vs. H₁: $\mu$ =950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours.NARREND

-{Rechargeable Batteries Narrative} Review the results of the previous questions. What is the effect of increasing the sample size on the value of $\beta$ ?

Question 41

A one-tail p-value is two times the size of a two-tail test.

Question 42

The larger the p-value, the more likely one is to reject the null hypothesis.

Question 43

In a one-tail test, the p-value is found to be equal to 0.054. If the test had been two-tail, then the p-value would have been 0.027.

Question 44

If we reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true.

Question 45

For a given level of significance, if the sample size is increased, the probability of committing a Type II error will decrease.

Question 46

The critical values will bound the rejection and non-rejection regions for the null hypothesis.

Question 47

If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level.

Question 48

If a null hypothesis is rejected at the 0.05 level of significance, it must be rejected at the 0.025 level.

Question 49

The p-value of a test is the smallest $\alpha$ at which the null hypothesis can be rejected.

Question 50

In order to determine the p-value, it is necessary to know the level of significance.

Question 51

A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence interval goes from 78.21 to 87.64. If the same sample had been used to test the null hypothesis that the mean of the population differs from 90, the null hypothesis could be rejected at a level of significance of 0.05.

Question 52

If we do not reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the null hypothesis is true.

Question 53

A p-value is a probability, and must be between 0 and 1.

Question 54

If your p-value is greater than 0.900 you should reject H₀ at the 0.10 level.

Question 55

A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence interval goes from 10.89 to 13.21. If the same sample had been used to test H₀: $\mu$ = 12 vs. H₁: $\mu$$\neq$ 12, H₀ could not be rejected at the 0.05 level.

Question 56

The p-value is the probability that the null hypothesis is true.

Question 57

Using the confidence interval when conducting a two-tail test for the population mean $\mu$ , we do not reject the null hypothesis if the hypothesized value for $\mu$ falls between the lower and upper confidence limits.

Question 58

A one-tail test for the population mean $\mu$ produces a test-statistic z = -0.75. The p-value associated with the test is 0.7734.

Question 59

For a given level of significance, if the sample size is increased, the probability of committing a Type I error will decrease.

Question 60

A two-tail test for the population mean $\mu$ produces a test-statistic z = 1.89. The p-value associated with the test is 0.0588.

Question 61

Suppose that we reject a null hypothesis at the 0.05 level of significance. Then for which of the following $\alpha$ -values do we also reject the null hypothesis?

A)0.06
B)0.04
C)0.03
D)0.02

Question 62

In a two-tail test for the population mean, if the null hypothesis is rejected when the alternative hypothesis is true:

A)a Type I error is committed.
B)a Type II error is committed.
C)a correct decision is made.
D)a one-tail test should be used instead of a two-tail test.

Question 63

In testing the hypotheses H₀: $\mu$ = 75 vs. H₁: $\mu$ < 75, if the value of the test statistic z equals -2.42, then the p-value is:

A)0.5078
B)2.4200
C)0.9922
D)0.0078

Question 64

In testing the hypothesis H₀: $\mu$ = 100 vs. H₁: $\mu$ > 100, the p-value is found to be 0.074, and the sample mean is 105. Which of the following statements is true?

A)The probability of observing a sample mean at least as large as 105 from a population whose mean is 100 is 0.074.
B)The probability of observing a sample mean smaller than 105 from a population whose mean is 100 is 0.074.
C)The probability that the population mean is larger than 100 is 0.074.
D)None of these choices.

Question 65

The critical values z _$\alpha$ or z _{$\alpha$ / 2} are the boundary values for:

A)the rejection region(s).
B)the level of significance.
C)Type I error.
D)Type II error.

Question 66

For a two-tail test, the null hypothesis will be rejected at the 0.05 level of significance if the value of the standardized test statistic z is:

A)smaller than 1.96 or greater than -1.96
B)greater than -1.96 or smaller than 1.96
C)smaller than -1.96 or greater than 1.96
D)greater than 1.645 or less than -1.645

Question 67

If we do not reject the null hypothesis, we conclude that:

A)there is enough statistical evidence to infer that the alternative hypothesis is true.
B)there is not enough statistical evidence to infer that the alternative hypothesis is true.
C)there is enough statistical evidence to infer that the null hypothesis is true.
D)there is not enough statistical evidence to infer that the null hypothesis is true.

Question 68

If we reject the null hypothesis, we conclude that:

A)there is enough statistical evidence to infer that the alternative hypothesis is true.
B)there is not enough statistical evidence to infer that the alternative hypothesis is true.
C)there is enough statistical evidence to infer that the null hypothesis is true.
D)there is not enough statistical evidence to infer that the null hypothesis is true.

Question 69

Statisticians can translate p-values into several descriptive terms. Suppose you typically reject H₀ at level 0.05. Which of the following statements is correct?

A)If the p-value < 0.001, there is overwhelming evidence to infer that the alternative hypothesis is true.
B)If 0.01 < p-value < 0.05, there is evidence to infer that the alternative hypothesis is true.
C)If p-value > 0.10, there is no evidence to infer that the alternative hypothesis is true.
D)All of these choices are true.

Question 70

If a hypothesis is not rejected at the 0.10 level of significance, it:

A)must be rejected at the 0.05 level.
B)may be rejected at the 0.05 level.
C)will not be rejected at the 0.05 level.
D)must be rejected at the 0.025 level.

Question 71

In order to determine the p-value, which of the following is not needed?

A)The level of significance.
B)Whether the test is one-tail or two-tail.
C)The value of the test statistic.
D)All of these choices are true.

Question 72

In testing the hypotheses H₀: $\mu$ = 800 vs. H₁: $\mu$ $\neq$ 800, if the value of the test statistic equals 1.75, then the p-value is:

A)0.0401
B)0.0802
C)0.4599
D)0.9599

Question 73

If a hypothesis is rejected at the 0.025 level of significance, it:

A)must be rejected at any level.
B)must be rejected at the 0.01 level.
C)must not be rejected at the 0.01 level.
D)may or may not be rejected at the 0.01 level.

Question 74

Suppose that in a certain hypothesis test the null hypothesis is rejected at the .10 level; it is also rejected at the .05 level; however it cannot be rejected at the .01 level. The most accurate statement that can be made about the p-value for this test is that:

A)p-value = 0.01.
B)p-value = 0.10.
C)0.01 < p-value < 0.05.
D)0.05 < p-value < 0.10.

Question 75

In a two-tail test for the population mean, the null hypothesis will be rejected at $\alpha$ level of significance if the value of the standardized test statistic z is such that:

A)z > z _$\alpha$
B)z < -z _$\alpha$
C)-z _$\alpha$ < z < z _$\alpha$
D)| z | > z _{$\alpha$ / 2}

Question 76

Which of the following p-values will lead us to reject the null hypothesis if the level of significance equals 0.05?

A)0.150
B)0.100
C)0.051
D)0.025

Question 77

In testing the hypotheses H₀: $\mu$ = 50 vs. H₁: $\mu$$\neq$ 50, the following information is known: n = 64, $\bar { x }$ = 53.5, and $\sigma$ = 10. The standardized test statistic z equals:

A)1.96
B)-2.80
C)2.80
D)-1.96

Question 78

In a one-tail test, the p-value is found to be equal to 0.068. If the test had been two-tail, the p-value would have been:

A)0.932
B)0.466
C)0.034
D)0.136

Question 79

If the value of the sample mean $\bar { x }$ is close enough to the hypothesized value If the value of the sample mean $\mu$ ₀ of the population mean If the value of the sample mean , then:

A)the value of If the value of the sample mean If the value of the sample mean $\mu$ ₀ is definitely correct.
B)the value of $\mu$ ₀ is definitely wrong.
C)we reject the null hypothesis.
D)we cannot reject the null hypothesis.

Question 80

Using a confidence interval when conducting a two-tail test for $\mu$ , we do not reject H₀ if the hypothesized value for $\mu$ :

A)is to the left of the lower confidence limit (LCL).
B)is to the right of the upper confidence limit (UCL).
C)falls between the LCL and UCL.
D)falls in the rejection region.