Deck 10: Hypothesis Tests Involving a Sample Mean or Proportion

Question 1

Reducing the probability of a Type I error also reduces the probability of a Type II error.

Answer 1

False

Question 2

Confidence intervals and hypothesis tests are both concerned with using sample information to make a statement about the unknown value of a population parameter.Thus,it is not surprising that their results are related.

Answer 2

True

Question 3

The p-value is the lowest possible level of significance at which the alternative hypothesis can be rejected.

Answer 3

False

Question 4

A Type II error is the probability of rejecting a true null hypothesis.

Question 5

The z-test is the appropriate test statistic for any population distribution with a small sample size and an unknown population standard deviation.

Question 6

A Type I error is the probability of failing to reject a false null hypothesis.

Question 7

If the p-value for a particular hypothesis test is 0.03 with $\alpha$ = 0.05,we do not reject the null hypothesis.

Question 8

In conducting a hypothesis test,we can choose any significance level we desire.In practice,however,levels of 0.10,0.05,and 0.01 tend to be most common.

Question 9

For a two-tail hypothesis test,the level of significance is the sum of both tail areas.

Question 10

In testing H0: $\mu \leq$ 120; H1: $\mu$ > 120 at the $\alpha$ = .05 level of significance,the null hypothesis is rejected.We must reach the same conclusion based on a confidence interval at the 95% confidence level.

Question 11

The rejection and non-rejection regions can be stated as a decision rule specifying the conclusion to be reached for a given outcome of the test.

Question 12

Directional assertions lead to what is called a two-tail test.

Question 13

In conducting a hypothesis test about the population mean $\mu$ ,if the population is truly or approximately normally distributed with $\text { ? }$ unknown,then the test statistic will be z,corresponding to the normal distribution.

Question 14

A directional claim or assertion states that a population parameter is greater than (>),at least,( $\ge$ ),no more than( $\le$ ),or less than (<),some quantity.

Question 15

The p-value of a test is the level of significance,where the observed value of the test statistic is exactly the same as a critical value for that level.

Question 16

A non-directional assertion involves what are called one-tail tests.

Question 17

The conclusion from a non-directional hypothesis test for a population mean at the $\alpha$ level of significance will be the same as the conclusion based on a confidence interval at the 100 (1 - $\alpha$ )% confidence level.

Question 18

Failure to reject a null hypothesis constitutes proof that it is true.

Question 19

The probability of Type I error is referred to as the significance level of the test.

Question 20

A non-directional claim or assertion states that a population parameter is equal to some quantity.

Question 21

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The decision rule is: Reject H₀ if the calculated z > 2.33.

Question 22

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The correct hypotheses are:
H₀: $\mu$ = 20,H₁: $\mu$ > 20

Question 23

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The decision rule is: Reject H₀ if the calculated t < -2.405.

Question 24

If a test is carried out at a specified significance level (e.g., $\alpha$ = .05),using a larger sample size will change the decision rule but will not change $\alpha$ .This is because $\alpha$ has been decided upon in advance.

Question 25

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The decision is: Because the test statistic t = 2.81 is greater than 2.33,reject H0.At the 0.01 level,the sample data suggest that the average P/E ratio for the stocks is less than 20.

Question 26

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The decision is: Because the test statistic t = -2.83 is not less than -2.33,fail to reject H0.At the 0.01 level,the sample data suggest that the average P/E ratio for the stocks is not less than 20.

Question 27

In a criminal trial where the null hypothesis states that the defendant is innocent,a Type II error is made when:

A) a guilty defendant is found not guilty.
B) an innocent person is found guilty.
C) a guilty defendant is found guilty.
D) an innocent person is found not guilty.

Question 28

A Type II error is defined as:

A) rejecting a true null hypothesis.
B) rejecting a false null hypothesis.
C) failing to reject a true null hypothesis.
D) failing to reject a false null hypothesis.

Question 29

If a hypothesis is rejected at the .025 level of significance,it:

A) must be rejected at any level.
B) must be rejected at the .01 level.
C) must not be rejected at the .01 level.
D) may be rejected or not rejected at the .01 level.

Question 30

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The correct hypotheses are:
H₀: $\mu$ = 20,H₁: $\mu$ =-20.

Question 31

If a test is carried out at a specified significance level (e.g., $\alpha$ = .05),using a larger sample size will reduce the value of $\beta$ ,the probability of making a Type II error.

Question 32

A Type I error occurs when we:

A) reject a false null hypothesis.
B) reject a true null hypothesis.
C) don't reject a false null hypothesis.
D) don't reject a true null hypothesis.

Question 33

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The decision rule is: Reject H₀ if the calculated t > 2.68 or if t < -2.68.

Question 34

Reducing the probability of a Type II error also reduces the probability of a Type I error.

Question 35

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The correct hypotheses are:
H₀: $\mu \geq$ 20,H₁: $\mu$ < 20.

Question 36

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

Question 37

In a criminal trial where the null hypothesis states that the defendant is innocent,a Type I error is made when:

A) a guilty defendant is found not guilty.
B) an innocent person is found guilty.
C) a guilty defendant is found guilty.
D) an innocent person is found not guilty.

Question 38

Suppose the underlying distribution is known to be normal,or that the sample size is at least thirty (30).The t-test is the appropriate test statistic when the sample standard deviation is used to estimate the population standard deviation.

Question 39

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance.
The decision is: Because the test statistic t = -2.81 falls in the rejection region,reject H0.At the 0.01 level,there is enough statistical evidence to conclude that the alternative hypothesis is true and that the average P/E ratio for the stocks is less than 20.

Question 40

Which of the following statements is not true?

A) The probability of making a Type I error is represented by the Greek letter.
B) The probability of making a Type II error is represented by the Greek letter.
C) The probability of making a Type II error is referred to as the level of significance.
D) For a given sample,reducing the probability of a Type I error increases the probability of a Type II error.
E) A Type I error can only occur if we reject the null hypothesis.

Question 41

The p-value of a test is:

A) the smallestat which the null hypothesis can be rejected.
B) the largestat which the null hypothesis can be rejected.
C) the smallestat which the null hypothesis cannot be rejected.
D) all of the above.

Question 42

A mortgage broker is offering home mortgages at a rate of 9.5% but is fearful that this value is higher than many others are charging.A sample of 40 mortgages filed in the county court house shows an average of 9.25% with a standard deviation of 8.61%.Does this sample indicate a smaller average? Use $\alpha$ = 0.05 and assume a normally distributed population.

A) Yes,the test statistic falls in the rejection region.
B) No,the test statistic falls in the acceptance region.
C) Yes,because the test statistic is greater than -1.645.
D) No,because the test statistic is -1.85 and falls in the rejection region.
E) Yes,because the sample mean of 9.25 is below 9.5.

Question 43

Consider testing the hypotheses H₀: $\mu$ = 50 vs.H₁: $\mu \neq$ 50. If n = 64, $\bar { x }$ = 53.5,and $\text { ? }$ = 10,then the value of the test statistic is:

A) z = 1.96.
B) t = 1.64.
C) z = 2.80.
D) t = 1.96.
E) z = 1.64

Question 44

Based on sample data,the 90% confidence interval limits for the population mean are 170.86 and 195.42.If the 10% level of significance was used in testing the hypothesesH₀: $\mu$ = 201,H₁: $\mu \neq$ 201,the null hypothesis:

A) would be rejected.
B) would not be rejected.
C) would have to be revised.
D) none of these.

Question 45

The power of a test is the probability of making:

A) a correct decision when the null hypothesis is false.
B) a correct decision when the null hypothesis is true.
C) an incorrect decision when the null hypothesis is false.
D) an incorrect decision when the null hypothesis is true.

Question 46

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) the test is statistically insignificant at whatever level of significance the test was conducted.

Question 47

H₀: ,H₁: ; n > 30 and = 0.01
Find

Question 48

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

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 49

Which of the following p-values will lead us to reject the null hypothesis if $\alpha$ = 0.05?

A) 0.15
B) 0.10
C) 0.05
D) 0.025

Question 50

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 or two tail
C) The value of the test statistic
D) All of these are needed.

Question 51

A spouse stated that the average amount of money spent on Christmas gifts for immediate family members is above $1200.The correct set of hypotheses is:

A) H₀:= 200,H₁:< 1200
B) H₀:> 1200,H₁:= 1200
C) H₀:= 1200,H₁:> 1200
D) H₀:< 1200,H₁:= 1200

Question 52

NARRBEGIN: Sermons
My daughter and I have argued about the average length of our preacher's sermons on Sunday morning.Despite my arguments,she thinks that the sermons are more than twenty minutes (over twenty minutes is important to her especially when soccer was playing on TV).For one year she has randomly selected 12 Sundays and found the average time of 26.42 minutes with a standard deviation of 6.69 minutes.Assuming that the population is normally distributed and using a 0.05 level of significance,is my daughter correct in thinking that the average length of sermons is more than 20 minutes?
What is the value of test statistic t?

Question 53

In testing the hypotheses H₀: $\pi$ = 0.40,H₁: $\pi$ > 0.40 at the 5% significance level,if the sample proportion is .45,and the standard error of the sample proportion is .035,the appropriate conclusion would be:

A) to reject H₀.
B) not to reject H₀.
C) to reject H₁.
D) to reject both H₀ and H₁.

Question 54

Consider testing the hypothesis H₀: $\mu$ = 800 vs.H₁: $\mu \neq$ 800. If the value of the test statistic z equals 1.75,then the p-value is:

A) 0.0401.
B) 0.0802.
C) 0.4599.
D) 0.9198.

Question 55

A professor of math refutes the claim that the average student spends 3 hours studying for the midterm exam.Which hypothesis is used to test the claim?

A) $H _ { 0 } : \mu \neq 3 , H _ { 1 } : \mu > 3$
B) $H _ { 0 } : \mu = 3 , H _ { 1 } : \mu \neq 3$
C) $H _ { 0 } : \mu \neq 3 , H _ { 1 } : \mu = 3$
D) $H _ { 0 } : \mu = 3 , H _ { 1 } : \mu < 3$

Question 56

In a one-tail test,the p-value is found to be equal to .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 57

Suppose that we reject a null hypothesis at the .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 58

In a one-tail test for the population mean,if the null hypothesis is not rejected when the alternative hypothesis is true then:

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

Question 59

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 $\mu$ :

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

Question 60

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) the test is statistically insignificant at whatever level of significance the test was conducted.

Question 61

A researcher wants to carry out a hypothesis test involving the mean for a sample of size n = 15.He does not know the true value of the population standard deviation,but is reasonably sure that the underlying population is approximately normally distributed.Should he use a z-test or a t-test in carrying out the analysis? Why?

Question 62

Merit
In order to receive a merit raise a computer clerk must average less than 15 minutes per order form processed.An across-the-board raise was given if the average is 15 minutes or more.The supervisor randomly selects 36 order forms and carefully times the length of time required to enter all of the data on each form.The significance level = 0.02.The average time to enter the 36 order forms is 14.36 minutes,and the population standard deviation is 3.03 minutes with a normal distribution.The test question is: Should a merit raise be given?

-State the null and alternative hypotheses.

Question 63

NARRBEGIN: Tail graph
Using the above graph,choose the answers that indicate a left-tail test:
Area 1: ____________________
Area 2: ____________________
Area 3: ____________________

Question 64

It has been claimed that no more than 3% of the units coming off an assembly line are defective.Formulate a null hypothesis and an alternative hypothesis for this situation.
H₀: ____________________
H₁: ____________________

Question 65

What standard error will be associated with this test?

Question 66

NARRBEGIN: Merit
In order to receive a merit raise a computer clerk must average less than 15 minutes per order form processed.An across-the-board raise was given if the average is 15 minutes or more.The supervisor randomly selects 36 order forms and carefully times the length of time required to enter all of the data on each form.The significance level = 0.02.The average time to enter the 36 order forms is 14.36 minutes,and the population standard deviation is 3.03 minutes with a normal distribution.The test question is: Should a merit raise be given?
State the decision rule in terms of the critical value as well as the p-value.

Question 67

The ____________________ is appropriate for hypothesis tests in which the sample standard deviation(s)is used in estimating the value of the population standard deviation ( ).

Question 68

Calculate the probability of a Type II error for the following test of hypothesis H₀: = 50 versus H₁: > 50 given that = 55, =.05, = 10,and n = 16.

Question 69

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

Question 70

Determine the power for the following test of hypothesis,H₀: = 950 vs.H₁: 950,given that = 1000, = 0.10, = 200,and n = 25.

Question 71

It has been claimed that no more than 3% of the units coming off an assembly line are defective.Will this test be a one-tail - left,a one tail - right,or a two-tail? Explain your answer.

Question 72

NARRBEGIN: Soft drink
A soft drink dispenser is set to automatically dispense drinks and ice in 8 oz.cups.The machine is set to fill the cups with 7 1/4 oz.so that the liquid will not run over the top of the cup.The company is getting complaints that the machine is filling the cups too full and spilling over.When the machine is operating properly,only 1% of the cups will not be filled properly.The production manager conducted a one-tail hypothesis test at the 0.01 level to check the machine.A sample of 50 fills was measured.
What is the critical value of z?

Question 73

A long-haul truck driver is eligible for a year-end bonus if the average miles driven per day are more than 375 miles per eight-hour day.To determine whether or not a driver is to receive a year-end bonus,a random sample of 36 days from the driver's logbook is taken and the sample mean is computed.The hypotheses are: H₀: 375 miles (no bonus)vs.H₁: > 375 miles (eligible for bonus).If the population standard deviation = 70 miles and the true mean is 405,what is the power of the test at = 0.05?

Question 74

For the critical z,identify the sample proportion that this z-value represents.
p = ____________________

Question 75

A Type II error is the probability of failing to reject a ____________________.It is represented by the Greek letter ____________________.

Question 76

A Type I error is represented by the Greek letter ____________________,which represents the probability of rejecting a ____________________.

Question 77

NARRBEGIN: Tail graph
Using the above graph,choose the answers that indicate a two-tail test:
Area 1: ____________________
Area 2: ____________________
Area 3: ____________________

Question 78

NARRBEGIN: Tail graph
Using the above graph,choose the answers that indicate a right-tail test:
Area 1: ____________________
Area 2: ____________________
Area 3: ____________________

Question 79

In order to receive a merit raise a computer clerk must average less than 15 minutes per order form processed. An across-the-board raise was given if the average is 15 minutes or more. The supervisor randomly selects 36 order forms and carefully times the length of time required to enter all of the data on each form. The significance level = 0.02. The average time to enter the 36 order forms is 14.36 minutes, and the population standard deviation is 3.03 minutes with a normal distribution. The test question is: Should a merit raise be given?
What is your conclusion?
Appropriate test: ____________________
Test statistic = ____________________
Conclusion: ____________________

Question 80

NARRBEGIN: Bank
A bank holds the mortgage on a motel which the motel owner says has an average daily revenue of at least $2048.In the event the average is not at least $2048,the bank will ask the motel owner for additional collateral for the loan as protection against loan default.A loan officer is aware that the daily average revenue is normally distributed and randomly selects 20 days during the last six months.The sample has a mean of $2003 and standard deviation of $144.00.At the 0.01 level,is the bank loan financially sound (should the bank ask for additional collateral)?
What is the value of test statistic t?