COMPREHENSIVE TUTORIAL _IV



COMPREHENSIVE TUTORIAL _V

Testing of Hypotheses: Basic Concepts, Type I and Type II Errors, One Tailed and Two Tailed Tests, One Sample Tests, Hypothesis Testing of Means when Population Standard Derivation is Known and when Unknown, Hypothesis Testing of Proportions for Large Samples. Two Sample Tests for Equality of Means for Large and Small Samples, Equality of Means for Dependant Samples, Difference between Proportions for Large Samples.

1. Explain the concept of Null Hypothesis & Alternative Hypothesis.

2. Define Type I error & Type II error

3. What does level of significance represent ?

4. What is the power of a test?

5. In hypothesis testing, the hypothesis tentatively assumed to be true is

a. the alternative hypothesis

b. the null hypothesis

c. either the null or the alternative

d. None of these alternatives is correct.

Answer: b

6. In hypothesis testing if the null hypothesis is rejected,

a. no conclusions can be drawn from the test

b. the alternative hypothesis is true

c. the data must have been accumulated incorrectly

d. the sample size has been too small

Answer: b

7. The level of significance is the

a. maximum allowable probability of Type II error

b. maximum allowable probability of Type I error

c. same as the confidence coefficient

d. same as the p-value

Answer: b

8. A Type II error is committed when

a. a true alternative hypothesis is mistakenly rejected

b. a true null hypothesis is mistakenly rejected

c. the sample size has been too small

d. not enough information has been available

Answer: a

9. The error of rejecting a true null hypothesis is

a. a Type I error

b. a Type II error

c. is the same as (

d. committed when not enough information is available

Answer: a

10. The level of significance

a. can be any positive value

b. can be any value

c. is (1 - confidence level)

d. can be any value between -1.96 to 1.96

Answer: c

11. The probability of making a Type I error is denoted by

a. (

b. (

c. 1 - (

d. 1 - (

Answer: a

12. The probability of making a Type II error is denoted by

a. (

b. (

c. 1 - (

d. 1 - (

Answer: b

13. When the following hypotheses are being tested at a level of significance of (

H0: ( ( 100

Ha: ( < 100

the null hypothesis will be rejected if the test statistic Z is

a. > Z(

b. > Z(

c. < -Z(

d. < 100

Answer: c

14. When the p-value is used for hypothesis testing, the null hypothesis is rejected if

a. p-value < (

b. ( < p-value

c. p-value > (

d. p-value = (

Answer: a

15. In order to test the following hypotheses at ( level of significance

H0: ( ( 100

Ha: ( > 100

the null hypothesis will be rejected if the test statistic Z is

a. > Z(

b. < Z(

c. < -Z(

d. < 100

Answer: a

16. In the hypothesis testing procedure, ( is

a. the level of significance

b. the critical value

c. the confidence level

d. 1 - level of significance

Answer: a

17. If a hypothesis test leads to the rejection of the null hypothesis

a. a Type II error must have been committed

b. a Type II error may have been committed

c. a Type I error must have been committed

d. a Type I error may have been committed

Answer: d

18. Your investment executive claims that the average yearly rate of return on the stocks she recommends is at least 10.0%. You plan on taking a sample to test her claim. The correct set of hypotheses is

a. H0: ( < 10.0% Ha: ( ( 10.0%

b. H0: ( ( 10.0% Ha: ( > 10.0%

c. H0: ( > 10.0% Ha: ( ( 10.0%

d. H0: ( ( 10.0% Ha: ( < 10.0%

Answer: d

19. A weatherman stated that the average temperature during July in Chattanooga is more than 80 degrees. A sample of 32 Julys is taken. The correct set of hypotheses is

a. H0: ( ( 80 Ha: ( ( 80

b. H0: ( ( 80 Ha: ( > 80

c. H0: ( ( 80 Ha: ( = 80

d. H0: ( < 80 Ha: ( > 80

Answer: b

20. A student believes that the average grade on the final examination in statistics is at least 85. She plans on taking a sample to test her belief. The correct set of hypotheses is

a. H0: ( < 85 Ha: ( ( 85

b. H0: ( ( 85 Ha: ( > 85

c. H0: ( ( 85 Ha: ( < 85

d. H0: ( > 85 Ha: ( ( 85

Answer: c

21. The school's newspaper reported that the proportion of students majoring in business is at least 30%. You plan on taking a sample to test the newspaper's claim. The correct set of hypotheses is

a. H0: P < 0.30 Ha: P ( 0.30

b. H0: P ( 0.30 Ha: P > 0.30

c. H0: P ( 0.30 Ha: P < 0.30

d. H0: P > 0.30 Ha: P ( 0.30

Answer: c

22. The average life expectancy of tires produced by the Whitney Tire Company has been 40,000 miles. Management believes that due to a new production process, the life expectancy of their tires has increased. In order to test the validity of their belief, the correct set of hypotheses is

a. H0: ( < 40,000 Ha: ( ( 40,000

b. H0: ( ( 40,000 Ha: ( > 40,000

c. H0: ( > 40,000 Ha: ( ( 40,000

d. H0: ( ( 40,000 Ha: ( < 40,000

Answer: b

23. In the past, 75% of the tourists who visited Chattanooga went to see Rock City. The management of Rock City recently undertook an extensive promotional campaign. They are interested in determining whether the promotional campaign actually increased the proportion of tourists visiting Rock City. The correct set of hypotheses is

a. H0: P > 0.75 Ha: P ( 0.75

b. H0: P < 0.75 Ha: P ( 0.75

c. H0: P ( 0.75 Ha: P < 0.75

d. H0: P ( 0.75 Ha: P > 0.75

Answer: d

24. A soft drink filling machine, when in perfect adjustment, fills the bottles with 12 ounces of soft drink. Any overfilling or under filling results in the shutdown and readjustment of the machine. To determine whether or not the machine is properly adjusted, the correct set of hypotheses is

a. H0: ( < 12 Ha: ( ( 12

b. H0: ( ( 12 Ha: ( > 12

c. H0: ( ( 12 Ha: ( = 12

d. H0: ( = 12 Ha: ( ( 12

Answer: d

25. The academic planner of a university thinks that at least 35% of the entire student body attends summer school. The correct set of hypotheses to test his belief is

a. H0: P > 0.35 Ha: P ( 0.35

b. H0: P ( 0.35 Ha: P > 0.35

c. H0: P ( 0.35 Ha: P < 0.35

d. H0: P > 0.35 Ha: P ( 0.35

Answer: c

26. The manager of an automobile dealership is considering a new bonus plan in order to increase sales. Currently, the mean sales rate per salesperson is five automobiles per month. The correct set of hypotheses for testing the effect of the bonus plan is

a. H0: ( < 5 Ha: ( ( 5

b. H0: ( ( 5 Ha: ( > 5

c. H0: ( > 5 Ha: ( ( 5

d. H0: ( ( 5 Ha: ( < 5

Answer: b

27. If a hypothesis is rejected at the 5% level of significance, it

a. will always be rejected at the 1% level

b. will always be accepted at the 1% level

c. will never be tested at the 1% level

d. may be rejected or not rejected at the 1% level

Answer: d

28. If a hypothesis is not rejected at the 5% level of significance, it

a. will also not be rejected at the 1% level

b. will always be rejected at the 1% level

c. will sometimes be rejected at the 1% level

d. None of these alternatives is correct.

Answer: a

29. For a two-tailed test with a sample size of 40, the null hypothesis will not be rejected at the 5% level of significance if the standardized test statistic is

a. between -1.96 and 1.96

b. greater than 1.96

c. less than 1.645

d. greater than -1.645

Answer: a

30. Changing from using the normal distribution to using the t distribution in a hypothesis test,

a. will result in the rejection region being smaller

b. will result in the rejection region being larger

c. would have no effect on the rejection region

d. None of these alternatives is correct.

Answer: a

31. The probability of rejecting a false null hypothesis is equal to

a. 1 - (

b. 1 - (

c. (

d. (

Answer: b

32. If a hypothesis is rejected at 95% confidence, it

a. will always be accepted at 90% confidence

b. will always be rejected at 90% confidence

c. will sometimes be rejected at 90% confidence

d. None of these alternatives is correct.

Answer: b

33. The manager of a grocery store has taken a random sample of 100 customers. The average length of time it took the customers in the sample to check out was 3.1 minutes with a standard deviation of 0.5 minutes. We want to test to determine whether or not the mean waiting time of all customers is significantly more than 3 minutes .Formulate the hypothesis

34. A random sample of 100 people was taken. Eighty of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 75%. Formulate hypothesis

35. You are given the following information obtained from a random sample of 5 observations.

20 18 17 22 18

At 90% confidence, you want to determine whether or not the mean of the population from which this sample was taken is significantly less than 21. (Assume the population is normally distributed.)

a. State the null and the alternative hypotheses.

b. Compute the standard error of the mean.

c. Determine the test statistic.

d. Test to determine whether or not the mean of the population is significantly less than 21.

Answers:

a. H0: ( ( 21

Ha: ( < 21

b. 0.89

c. t = -2.25

d. Reject H0, the mean is significantly less than 21.

36. Consider the following hypothesis test:

Ho: p = 0.5

Ha: p ( 0.5

A sample of 800 provided a sample proportion of 0.58.

a. Using ( = 0.05, what is the rejection rule?

b. Determine the standard error of the proportion.

c. Compute the value of the test statistic z. What is your conclusion?

d. Determine the p-value.

Answers:

a. Reject H0 if z < -1.96 or if z > 1.96

b. 0.01767

c. 4.52; reject H0

d. zero

37. A new soft drink is being market tested. A sample of 400 individuals participated in the taste test and 80 indicated they like the taste.

a. At 95% confidence, test to determine if at least 22% of the population will like the new soft drink.

b. Determine the p-value.

Answers:

a. H0: p( 0.22

Ha: p < 0.22 Z = -1; therefore, do not reject H0

b. 0.1587

38. A sample of 81 account balances of a credit company showed an average balance of $1,200 with a standard deviation of $126. You want to determine if the mean of all account balances is significantly different from $1,150. Use a .05 level of significance.

Answer:

H0: ( = 1150

Ha: ( 1150 Z = 3.57; therefore, reject H0

39. From a population of cans of coffee marked "12 ounces," a sample of 25 cans is selected and the contents of each can are weighed. The sample revealed a mean of 11.8 ounces with a standard deviation of 0.5 ounces. Test to see if the mean of the population is at least 12 ounces. (Assume the population is normally distributed.) Use a .05 level of significance.

40. In the past the average age of employees of a large corporation has been 40 years. Recently, the company has been hiring older individuals. In order to determine whether there has been an increase in the average age of all the employees, a sample of 25 employees was selected. The average age in the sample was 45 years with a standard deviation of 5 years. Assume the distribution of the population is normal. Let ( = .05.

a. State the null and the alternative hypotheses.

b. Test to determine whether or not the mean age of all employees is significantly more than 40 years.

Answers:

a. H0: ( ( 40

Ha: ( > 40

b. t = 5; therefore, reject H0

41. A lathe is set to cut bars of steel into lengths of 6 centimeters. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 6 centimeters. A sample of 121 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 6.08 centimeters with a standard deviation of 0.44 centimeters. Determine whether or not the lathe is in perfect adjustment. Use a .05 level of significance.

Answer:

H0: ( = 6

Ha: ( ( 6 Z = 2; therefore, reject H0

42. Bastien, Inc. has been manufacturing small automobiles that have averaged 50 miles per gallon of gasoline in highway driving. The company has developed a more efficient engine for its small cars and now advertises that its new small cars average more than 50 miles per gallon in highway driving. An independent testing service road-tested 36 of the automobiles. The sample showed an average of 51.5 miles per gallon with a standard deviation of 6 miles per gallon.

a. With a 0.05 level of significance, test to determine whether or not the manufacturer's advertising campaign is legitimate.

b. What is the p-value associated with the sample results?

Answers:

a. H0: ( ( 50

Ha: ( > 50 Z = 1.5, therefore, do

43. Identify the null and alternative hypotheses for the following problems.

a. The manager of a restaurant believes that it takes a customer no more than 25 minutes to eat lunch.

b. Economists have stated that the marginal propensity to consume is at least 90¢ out of every dollar.

c. It has been stated that 75 out of every 100 people who go to the movies on Saturday night buy popcorn.

Answers:

a. H0: ( ( 25

Ha: ( > 25

b. H0: p ( 0.9

Ha: p < 0.9

c. H0: p = 0.75

Ha: p ( 0.75

44. A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days with a standard deviation of 5.6 days.

a. State the null and alternative hypotheses.

b. Using a standardized test statistic, test the null hypothesis at the 5% level of significance.

ANSWERS:

a. H0: ( ( 15

Ha: ( > 15

b. Do not reject H0, 1.5 < 1.645

45. A law enforcement agent believes that at least 88% of the drivers stopped on Saturday nights for speeding are under the influence of alcohol. A sample of 66 drivers who were stopped for speeding on a Saturday night was taken. Eighty percent of the drivers in the sample were under the influence of alcohol.

a. State the null and alternative hypotheses.

b. Using a standardized test statistic, test the hypothesis at the 1% level of significance.

c. Using a p-value, test the hypothesis at the 1% level of significance.

Answers:

a. H0: P ( 0.88

Ha: P < 0.88

b. Do not reject H0; -2 > -2.33

c. Do not reject H0; 0.0228 > 0.01

46. An automobile manufacturer stated that it will be willing to mass produce electric-powered cars if more than 30% of potential buyers indicate they will purchase the newly designed electric cars. In a sample of 500 potential buyers, 160 indicated that they would buy such a product. Should the manufacturer produce the new electric-powered cars? Use a 0.05 level of significance. (Hint: Ho: P ( 0.3)

Answer:

H0: P ( 0.3

Ha: P > 0.3

Z = 0.98; do not reject H0; no, the manufacturer should not produce the cars.

47. It is said that more males register to vote in a national election than females. A research organization selected a random sample of 300 registered voters and reported that 165 of the registered voters were male. Based on the sample results, can you conclude that more males registered to vote than females? Use a 0.05 level of significance. (Hint: Ho: P ( 0.5)

Answer:

H0: P ( 0.5

Ha: P > 0.5

Z = 1.78; reject H0; yes, more males than females registered to vote.

48. When each data value in one sample is matched with a corresponding data value in another sample, the samples are known as

a. corresponding samples

b. matched samples

c. independent samples

d. None of these alternatives is correct.

Answer: b

49. The following information was obtained from matched samples.

The daily production rates for a sample of workers before and after a training program are shown below.

Worker Before After

1 20 22

2 25 23

3 27 27

4 23 20

5 22 25

6 20 19

7 17 18

. The point estimate for the difference between the means of the two populations is

a. -1

b. -2

c. 0

d. 1

Answer: c

50. Refer to 49. The null hypothesis to be tested is H0: (d = 0. The test statistic is

a. -1.96

b. 1.96

c. 0

d. 1.645

Answer: c

51. Refer to 49. Based on the results the

a. null hypothesis should be rejected

b. null hypothesis should not be rejected

c. alternative hypothesis should be accepted

d. None of these alternatives is correct.

Answer: b

52. If we are interested in testing whether the mean of population 1 is smaller than the mean of population 2, the

a. null hypothesis should state (1 - (2 < 0

b. null hypothesis should state (1 - (2 > 0

c. alternative hypothesis should state (1 - (2 ( 0

d. alternative hypothesis should state (1 - (2 ( 0

Answer: d

53. If we are interested in testing whether the proportion of items in population 1 is larger than the proportion of items in population 2, the

a. null hypothesis should state P1 - P2 < 0

b. null hypothesis should state P1 - P2 ( 0

c. alternative hypothesis should state P1 - P2 ( 0

d. alternative hypothesis should state P1 - P2 ( 0

Answer: c

54. In order to estimate the difference between the average Miles per Gallon of two different models of automobiles, samples are taken and the following information is collected.

Model A Model B

Sample Size 50 55

Sample Mean 32 35

Sample Variance 9 10

a. At 95% confidence develop an interval estimate for the difference between the average Miles per Gallon for the two models.

b. Is there conclusive evidence to indicate that one model gets a higher MPG than the other? Why or why not? Explain.

Answers:

a. -4.18 to -1.82

b.Since the range of the interval is from negative to negative, it indicates that there is conclusive evidence (at 95%) that Model B has a larger mean

55. The management of Recover Fast Hospital (RFH) claims that the average length of stay in their hospital after a major surgery is less than the average length of stay at General Hospital (GH). The following data has been accumulated to test their claim. Assume the two populations are normally distributed and have equal variances.

RFH GH

Sample size 16 26

Mean (in days) 4 5

Std. Deviation 0.5 0.75

Test to see if the average length of stay in RFH is significantly less than the average length of stay in GH. Let ( = 0.05.

Answer:

Test statistic t = -4.716 < -1.684. Reject Ho, and conclude the length of stay is RFH is significantly less than GH.

56. In order to determine whether or not a driver's education course improves the scores on a driving exam, a sample of 6 students were given the exam before and after taking the course. The results are shown below.

Let d = Score After - Score Before.

Score Score

Student Before the Course After the Course

1 83 87

2 89 88

3 93 91

4 77 77

5 86 93

6 79 83

Use ( = 0.1 and test to see if taking the course actually increased scores on the driving exam.

Answer:

Test statistic t = 1.3912 0 t = 0.8; do not reject H0

60. The daily production rates for a sample of factory workers before and after a training program are shown below.

Worker Before After

1 6 9

2 10 12

3 9 10

4 8 11

5 7 9

At 95% confidence, test to see if the training program was effective. That is, did the training program actually increase the production rates?

Answer:

H0: (d ( 0

Ha: (d > 0 t = 5.8; reject H0

61. In a sample of 100 Republicans, 60 favored the President's anti-drug program. While in a sample of 150 Democrats, 84 favored his program. At 95% confidence, test to see if there is a significant difference in the proportions of the Democrats and the Republicans who favored the President's anti-drug program.

Answer:

H0: P1 - P2 = 0

Ha: P1 - P2 ( 0 Z = 0.625; do not reject H0

62. The following data presents the number of computer units sold per day by a sample of 6 salespersons before and after a bonus plan was implemented.

Salesperson Before After

1 3 6

2 7 5

3 6 6

4 8 7

5 7 8

6 9 8

At 95% confidence, test to see if the bonus plan was effective. That is, did the bonus plan actually increase sales?

Answer:

H0: ( ( 0

Ha: ( > 0 t = 0; do not reject H0

63. The office of records at a university has stated that the proportion of incoming female students who major in business has increased. A sample of female students taken several years ago is compared with a sample of female students this year. Results are summarized below. Has the proportion increased significantly? Test at alpha = .10.

Sample Size No. Majoring in Business

Previous Sample 250 50

Present Sample 300 69

Answer:

H0: P1 - P2 ( 0

Ha: P1 - P2 > 0 Z = 0.85; do not reject H0

64. Allied Corporation is trying to sell its new machines to Ajax. Allied claims that the machine will pay for itself since the time it takes to produce the product using the new machine is significantly less than the production time using the old machine. To test the claim, independent random samples were taken from both machines. You are given the following results.

New Machine Old Machine

Sample Mean 25 23

Sample Variance 27 7.56

Sample Size 45 36

a. State the null and alternative hypotheses for Allied Corporation's claim.

b. What are the point estimates for the mean and the standard deviation of the difference between the means of the two populations?

c. Use the standardized test statistic to test Allied Corporation's claim. Use a .05 level of significance.

d. What does the p-value equal? Use the p-value to test Allied Corporation's claim. Use a .05 level of significance.

e. As the statistical advisor to Ajax, would you recommend purchasing Allied's machine? Explain your answer.

ANSWERS:

a. H0: (1 - (2 ( 0

Ha: (1 - (2 > 0

b. 2 and .9

c. Reject H0; 2.222 > 1.645

d. p-value = 0.0132; reject H0; 0.0132 < 0.05

e. Ajax should not purchase Allied's machine since the null hypothesis was rejected.

65. A school administrator believes that there is no difference in the student dropout rate for schools located in his district and schools located in another district. A random sample of 25 schools in the administrator's district was taken. The student dropout rate of the schools in the sample was 24%. A random sample of 30 schools in the other district had a dropout rate of 27%.

a. Give a point estimate for the difference between the population proportions for the two districts.

b. Give a point estimate of the standard deviation for the difference between the population proportions for the two districts.

c. Construct a 99% confidence interval for the difference between the population proportions for the two districts.

d. State the null and alternative hypotheses.

e. Test the hypothesis stated in Part d at the 1% significance level.

f. What do you conclude?

Answers:

a. -0.03

b. 0.1178

c. -0.33 to 0.27

d. H0: P1 - P2 = 0

Ha: P1 - P2 ( 0

e. Do not reject H0; -2.58 < -0.249 < 2.58

f. There is not sufficient evidence to reject the administration's belief.

66. An insurance company selected samples of clients under 18 years of age and over 18 and recorded the number of accidents they had in the previous year. The results are shown below.

Under Age of 18 Over Age of 18

n1 = 400 n2 = 900

Number of accidents = 76 Number of accidents = 90

Test with ( = .05 to determine if the accident proportions differ between the two groups.

Answer:

H0: Pu - Po = 0

Ha: Pu - Po ( 0

Z = 4.5 > 1.96; reject Ho

67. A recent Time magazine reported the following information about workers in Germany and the United States:

United States Germany

Average length of workweek (hours) 42 38

Standard Deviation 5 6

Sample Size 600 700

At 95% confidence, test to determine whether or not there is a significant difference between the average workweek in the United States and the average workweek in Germany.

Answer:

H0: (U - (G = 0

Ha: (U - (G ( 0

Z = 13.1 > 1.96; reject Ho

68. A pharmaceutical firm has requested that a medical doctor observe whether or not a new version of a drug has actually shortened the length of time it takes for a patient to feel the benefits of the medication. It can be assumed that the reaction time for the drug is normally distributed and that the standard deviations of the reaction times are equal for both versions of the drug. The doctor has administered the older drug and its new version to independent random samples of his patients.

Old New

Version Version

Sample Size 15 10

Sample Mean 80 minutes 65 minutes

Sample Standard Deviation 10 minutes 14 minutes

a. Give the pooled estimate of the population variance.

b. Calculate the standard deviation of the sampling distribution for the difference between the population means.

c. What form should be used for the sampling distribution of the difference in means?

d. Give the 95% confidence interval estimate for the true difference in reaction time.

e. If the new version is more costly to manufacture than the former version, what advice should the doctor give to the pharmaceutical company? Use a .05 level of significance.

Answers:

a. 137.565

b. 4.788

c. t distribution with 23 degrees of freedom

d. 5.094 to 24.906

e. The null hypothesis of no improvement can be rejected at the 5% level

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