# Hypothesis Testing - Mean, Proportion & ANOVA

10.9 A problem with a telephone line that prevents a customer from receiving or making calls is disconcerting to both the customer and the telephone company. The file contains samples of 20 problems reported to two dif-ferent offices of a telephone company and the time to clear these problems ( in minutes) from the customers' lines:

Central Office I Time to Clear Problems (minutes)

1.48 1.75 0.78 2.85 0.52 1.60 4.15 3.97 1.48 3.10

1.02 0.53 0.93 1.60 0.80 1.05 6.32 3.93 5.45 0.97

Central Office II Time to Clear Problems (minutes)

7.55 3.75 0.10 1.10 0.60 0.52 3.30 2.10 0.58 4.02

3.75 0.65 1.92 0.60 1.53 4.23 0.08 1.48 1.65 0.72

a. Assuming that the population variances from both offices are equal, is there evidence of a difference in the mean waiting time between the two offices? ( Use )

b. Find the p- value in (a) and interpret its meaning.

c. What other assumption is necessary in (a)?

d. Assuming that the population variances from both offices are equal, construct and interpret a 95% confidence interval estimate of the difference between the population means in the two offices.

10.11 Digital cameras have taken over the majority of the point- and- shoot camera market. One of the important features of a camera is the battery life, as measured by the number of shots taken until the battery needs to be recharged. The file contains the battery life of 29 sub-compact cameras and 16 compact cameras (data extracted from " Digital Cameras," Consumer Reports, July 2009, pp. 28- 29).

a. Assuming that the population variances from both types of digital cameras are equal, is there evidence of a difference in the mean battery life between the two types of digital cameras (a=0.05)?

b. Determine the p-value in ( a) and interpret its meaning.

c. Assuming that the population variances from both types of digital cameras are equal, construct and interpret a 95% confidence interval estimate of the difference between the population mean battery life of the two types of digital cameras.

10.20 Nine experts rated two brands of Colombian coffee in a taste- testing experiment. A rating on a 7- point scale ( extremely unpleasing, extremely pleasing) is given for each of four characteristics: taste, 1 = 7 = extremely pleasing) is given for each of our characteristics: taste, aroma, richness, and acidity. The following data (stored in Coffee) display the summated ratings- accumulated over all four characteristics.

BRAND

EXPERT A B

C. C. 24 26

S. E. 27 27

E. G. 19 22

B. L. 24 27

C. M. 22 25

C. N. 26 27

G. N. 27 26

R. M. 25 27

P. V. 22 23

a. At the 0.05 level of significance, is there evidence of a difference in the mean summated ratings between the two brands?

b. What assumption is necessary about the population distribution in order to perform this test?

c. Determine the p- value in ( a) and interpret its meaning.

d. Construct and interpret a 95% confidence interval estimate of the difference in the mean summated ratings between the two brands.

10.31 Some people enjoy the anticipation of an upcoming product or event and prefer to pay in advance and delay the actual consumption/ delivery date. In other cases, people do not want a delay. An article in the Journal of Marketing Research reported on an experiment in which 50 individuals were told that they had just purchased a ticket to a concert and 50 were told that they had just purchased a personal dig-ital assistant ( PDA). The participants were then asked to indicate their preferences for attending the concert or receiving the PDA. Did they prefer tonight or tomorrow, or would they prefer to wait two to four weeks? The individuals were told to ignore their schedule constraints in order to better measure their willingness to delay the consumption/ delivery of their purchase. The following table gives partial results of the study:

Concert PDA

Tonight or tomorrow 28 47

Two to four weeks 22 3

Total 50 50

Source: Data adapted from O. Amir and D. Ariely, " Decisions by Rules: The Case of Unwillingness to Pay for Beneficial Delays," Journal of Marketing Research, February 2007, Vol. XLIV, pp. 142- 152.

a. What proportion of the participants would prefer delaying the date of the concert?

b. What proportion of the participants would prefer delaying receipt of a new PDA?

c. Using the 0.05 level of significance, is there evidence of a significant difference in the proportion willing to delay the date of the concert and the proportion willing to delay receipt of a new PDA?

10.35 Where people turn for news is different for various age groups (data extracted from P. Johnson, " Young People Turn to the Web for News," USA Today, March 23, 2006, p. 9D). Suppose that a study conducted on this issue was based on 200 respondents who were between the ages of 36 and 50 and 200 respondents who were above age 50. Of the 200 respondents who were between the ages of 36 and 50, 82 got their news primarily from newspapers. Of the 200 respondents who were above age 50, 104 got their news primarily from newspapers.

a. Is there evidence of a significant difference in the proportion that get their news primarily from newspapers between those respondents 36 to 50 years old and those above 50 years old? (Use )

b. Determine the p- value in ( a) and interpret its meaning. a = 0.05.

c. Construct and interpret a 95% confidence interval estimate for the difference between the population proportion of respondents who get their news primarily from newspapers between those respondents 36 to 50 years old and those above 50 years old.

10.61 Do male and female students study the same amount per week? In 2007, 58 sophomore business students were surveyed at a large university that has more than 1,000 sophomore business students each year. The file StudyTime contains the gender and the number of hours spent studying in a typical week for the sampled students.

a. At the 0.05 level of significance, is there a difference in the variance of the study time for male students and female students?

b. Using the results of ( a), which t test is appropriate for comparing the mean study time for male and female students?

c. At the 0.05 level of significance, conduct the test selected in ( b).

d. Write a short summary of your findings.

10.67 A hotel manager is concerned with increasing the return rate for hotel guests. One aspect of first impressions by guests relates to the time it takes to deliver the guest's luggage to the room after check- in to the hotel. A random sample of 20 deliveries on a particular day were selected in Wing A of the hotel, and a random sample of 20 deliveries were selected in Wing B. The results are stored in Luggage . Analyze the data and determine whether there is a difference in the mean delivery time in the two wings of the hotel. ( Use a=0.05.)

11.7 The Computer Anxiety Rating Scale (CARS) measures an individual's level of computer anxiety, on a scale from 20 ( no anxiety) to 100 ( highest level of anxiety). Researchers at Miami University administered CARS to 172 business students. One of the objectives of the study was to determine whether there are differences in the amount of computer anxiety experienced by students with different majors. They found the following:

Degrees of Sum of Mean

Source Freedom Squares Squares F

Among majors 5 3,172

Within majors 166 21,246

Total 171 24,418

Major n Mean

Marketing 19 44.37

Management 11 43.18

Other 14 42.21

Finance 45 41.80

Accountancy 36 37.56

MIS 47 32.21

Source: Extracted from T. Broome and D. Havelka, " Determinants of Computer Anxiety in Business Students," The Review of Business Information Systems, Spring 2002, 6( 2), pp. 9- 16.

a. Complete the ANOVA summary table.

b. At the 0.05 level of significance, is there evidence of a difference in the mean computer anxiety experienced by different majors?

c. If the results in ( b) indicate that it is appropriate, use the Tukey- Kramer procedure to determine which majors differ in mean computer anxiety. Discuss your findings.

11.9 A hospital conducted a study of the waiting time in its emergency room. The hospital has a main campus and three satellite locations. Management had a business objective of reducing waiting time for emergency room cases that did not require immediate attention. To study this, a random sample of 15 emergency room cases that did not require immediate attention at each location were selected on a particular day, and the waiting time (measured from check- in to when the patient was called into the clinic area) was measured. The results are stored in ERWaiting .

a. At the 0.05 level of significance, is there evidence of a difference in the mean waiting times in the four locations?

b. If appropriate, determine which locations differ in mean waiting time.

c. At the 0.05 level of significance, is there evidence of a difference in the variation in waiting time among the four locations?

11.13 A pet food company has a business objective of expanding its product line beyond its current kidney- and shrimp- based cat foods. The company developed two new products, one based on chicken livers and the other based on salmon. The company conducted an experiment to compare the two new products with its two existing ones, as well as a generic beef-based product sold in a supermarket chain. For the experiment, a sample of 50 cats from the population at a local animal shelter was selected. Ten cats were randomly assigned to each of the five products being tested. Each of the cats was then presented with 3 ounces of the selected food in a dish at feeding time. The researchers defined the variable to be measured as the number of ounces of food that the cat consumed within a 10- minute time interval that began when the filled dish was presented. The results for this experiment are summarized in the following table and stored in CatFood.

Kidney Shrimp Chicken Liver Salmon Beef

2.37 2.26 2.29 1.79 2.09

2.62 2.69 2.23 2.33 1.87

2.31 2.25 2.41 1.96 1.67

2.47 2.45 2.68 2.05 1.64

2.59 2.34 2.25 2.26 2.16

2.62 2.37 2.17 2.24 1.75

2.34 2.22 2.37 1.96 1.18

2.47 2.56 2.26 1.58 1.92

2.45 2.36 2.45 2.18 1.32

2.32 2.59 2.57 1.93 1.94

a. At the 0.05 level of significance, is there evidence of a difference in the mean amount of food eaten among the various products?

b. If appropriate, which products appear to differ significantly in the mean amount of food eaten?

c. At the 0.05 level of significance, is there evidence of a significant difference in the variation in the amount of food eaten among the various products?

d. What should the pet food company conclude? Fully describe the pet food company's options with respect to the products.

https://brainmass.com/statistics/analysis-of-variance/hypothesis-testing-mean-proportion-anova-372646

#### Solution Summary

The solution provides step by step method for the calculation of testing of hypothesis. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

Statistics: Cell phone battery, Lovastatin, bumper test, LCD projector

See data file attached.

Directions: You may include the statistical software output, but you must also include a well-written explanation of the findings. Be sure to answer the question asked in each problem, and explain why, with reference to your output. If you calculate the answers manually, be sure to show your work. I would prefer a Word document with your answers below each problem, but you may also submit an Excel document.

1: 10.20 A new cell phone battery is being considered as a replacement for the current one. Ten college student cell phone users are selected to try each battery in their usual mix of "talk" and "standby" and to record the number of hours until recharge was needed. (a) Do these results show that the new battery has significantly longer life at α = .05? State your hypotheses and show all steps clearly. (b) What is your decision on the null hypothesis? Is the decision close? (c) Are you convinced?

Cell Phone Battery Life Experiment (n = 10)

Participant New Battery Old Battery

Bob 45 52

May 41 34

Deno 53 40

Sri 40 38

Pat 43 38

Alexis 43 44

Scott 49 34

Aretha 39 45

Jen 41 28

Ben 43 33

2: 10.44: Does lovastatin (a cholesterol-lowering drug) reduce the risk of heart attack? In a Texas study, researchers gave lovastatin to 2,450 people and an inactive substitute to 2,370 people (average age 58). After 5 years, 58 of the lovastatin group had suffered a heart attack, compared with 97 for the inactive pill. (a) State the appropriate null and alternative hypotheses. (b) Interpret the results at α = .01. Do you reject the null hypothesis of no difference? (c) Is normality assured? Why or why not? (d) Is the difference large enough to be important?

3: 11.24 In a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with the results shown below. Research questions: (a) Are the mean crash damages the same or significantly different among these three vehicles? (b) If there is at least one significant difference among the groups, are there any significant differences between pairs of groups? What multiple comparison test did you use? Explain.

Crash Damage in Dollars

Goliath Varmint Weasel

1100 1290 1100

750 1400 1500

970 1390 1000

1000 1850 1250

850 1100 1920

Directions: For the following hypothesis tests, identify the null and alternative hypothesis, and the critical value. Then, calculate the test statistic, note the p value and make a decision on the null hypothesis. Please show your work if you calculated manually. If you used statistical software, please show output. Your p value will be approximate if you use manual calculation (i.e., less than .05) or exact (if you used statistical software).

1. Hypothesis test for the difference of population means: t test

A purchasing manager for a large university is investigating which brand of LCD projector to purchase to equip "smart" classrooms. Of major concern is the longevity of the light bulbs used in the projectors. The purchasing manager has narrowed down the choice of projector to two brands, Infocus and Proxima, and wishes to determine if there is any difference between the two brands in the mean lifetime of the bulbs used.

The purchasing manager obtained thirteen projectors of each brand for testing over the last several academic terms. The number of hours the bulbs lasted on each of the thirteen machines is given in the table.

Infocus Proxima

762 900

954 720

854 1132

935 631

861 690

908 1112

890 732

1012 754

964 1032

807 865

1013 953

798 754

923 778

Assume that the two populations of lifetimes are normally distributed and that the population variances are equal. Can we conclude, at the 0.01 level of significance, that there is a difference in the mean lifetime of the light bulbs in the two brand

2. Hypothesis test for the difference of population means: ANOVA

The manager of a computer software company wishes to study the number of hours senior executives spend at their desktop computers by type of industry. The manager selected a sample of five executives from each of three industries. At the .05 significance level, can she conclude there is a difference in the mean number of hours spent per week by industry?

Banking Retail Insurance

10 7.5 8

11 7 8.5

9 5 8

8 5 5

10 6 7

Answers:

1: State the null & alternative hypothesis:

2: Identify critical value

3: Calculate test statistic

(p value)

4: State your decision on H0

5: If you reject H0, identify groups that are significantly different from each other.

Answers:

1: State the null & alternative hypothesis:

2: Identify critical value

3: Calculate test statistic

(p value)

4: State your decision on H0