# Several ANOVA problems

1. Jointsoft is a great over-the-counter arthritis medication, but who will ever know about it? Unfortunately, many people with arthritis tend to be elderly and rather immobile, so advertisers of arthritis medications face limitations in ways to get their messages across. Currently, their best modes of advertisement are commercials on daytime TV, advertisements in select magazines, fliers in convalescent homes, and (believe it or not) advertisements on certain Web pages.

Marketing managers for Jointsoft are investigating these four modes of advertisement in four small communities (with a different mode of advertisement in each community). The marketing managers have selected days at random and are looking at the daily sales (in dollars) in each of the communities on each of these days. Here is what they have to work with:

Groups Sample Size Sample Mean Sample Variance

TV 37 554.5 2031.1

Magazines 37 575.3 2552.8

Fliers 37 561.8 3197.3

Web Pages 37 570.4 2697.9

Suppose that the marketing managers perform an ANOVA test to decide if there are differences in the mean daily sales arising from the four modes of advertisement. (So, they're assuming that the only difference among the four communities is the mode of advertisement used in it.) An ANOVA uses the test statistic

Variation between the samples .

Variation within the samples

For the information in the chart above, .

Give the p-value corresponding to this value of the F statistic. Round at least 3 decimal places.

Using the 0.05 level of significance, can the marketing managers conclude thatq the mean daily sales arising from at least one of the modes of advertisement differs from the others? YES or NO

2. The General Social Survey is an annual survey given to a random selection of about adults in the United States. Among the many questions asked are "What is the highest level of education you've completed?" and "If you're employed full-time, how many hours do you spend working at your job during a typical week?"

In a recent year, respondents answered both questions. The summary statistics are given in the chart below. (The sample data consist of the times, in hours per week, that were given by the respondents.)

Groups Sample Size Sample Mean Sample Variance

Less than h.s. 257 41.8 108.2

High School 298 41.2 99.7

Bachelor's 254 42.5 99.6

Graduate 273 43.5 115.9

To decide if there are any differences in the mean hours per week worked by these different groups, we can perform an ANOVA. An ANOVA uses the test statistic

Variation between the samples .

Variation within the samples

For the data from the survey, .

Give the p-value corresponding to this value of the F-statistic. Round to at least 3 decimal places.

From the survey data, can we conclude that at least one of the groups differs significantly from the others in mean hours worked in a typical week? Use the 0.05 level of significance. YES or NO

3. Cris Turlock owns and manages a small business in San Francisco, California. The business provides breakfast and brunch food, via carts parked along sidewalks, to people in the business district of the city.

Being an experienced businessperson, Cris provides incentives for the salespeople operating the food carts. This year, she plans to offer monetary bonuses to her salespeople based on their individual mean daily sales. Her first task is to see if there is a significant difference in the mean daily sales among the different salespeople. She chooses a "sample" of days for each salesperson and records the sales for each day. She then runs a one-way, independent-samples ANOVA test to determine whether or not she can conclude that at least one salesperson's performances is significantly different from the others. (Otherwise, she'll split the bonuses evenly among all the salespeople.) In her ANOVA, the "groups" are the different salespeople, and the variable of interest is the daily sales amount, in dollars.

Below is an ANOVA table summarizing Cris' ANOVA. Fill in the missing cell of the table (round your answer to at least two decimal places), and then answer the questions about the ANOVA.

Source of Variation Degrees of Freedom Sum of Squares Mean Squares F Statistics

Treatments (Between Groups) 5 40558.8 8111.8 ____?____

Error (Within Groups) 638 1543099.8 2418.7

Total

643 1583658.6

a.) How many sales people were included in the analysis?

b.) For the ANOVA test, it is assumed that the population variance of daily sales is the same of each salesperson. What is an unbiased estimate of this common population variance based on the sample variance?

c.) What is the p-value corresponding to the F statistics for the ANOVA test? Round your answer to atleast three decimal places.

d.) Can we conclude, using the 0.05 level of significance, that at least one salesperson's mean daily sales is significantly different from that of the others? YES or NO

4. Emma's On-the-Go, a large convenience store, has to decide where in the store to put its magazine rack. The manager at Emma's experiments with a selection of different locations, choosing a sample of days at each location. Each day, the manager records the amount of money brought in from the sale of magazines.

It's possible to test whether there is a difference in the mean daily sales for the different locations by doing a one-way, independent-samples ANOVA test. The variable of interest is the daily sales, in dollars, from magazines at Emma's. In the ANOVA test, the "groups" are the different locations, and the "samples" are the daily magazine sales actually examined by the manager.

The following ANOVA table gives a summary of such an ANOVA test. Fill in the missing cell in the table (rounded to two decimal places), and then answer the questions.

Source of Variation Degrees of Freedom Sum of Squares Mean Squares F Statistics

Treatments (Between Groups) 5 4305.9 861.2 ____?____

Error (Within Groups) 234 103669.8 443

Total

239 107875.7

e.) How many locations were looked at by the manager?

f.) For the ANOVA test, it is assumed that the variance is the same for each population of daily sales (that is, for the population of daily sales for each location).What is an unbiased estimate of this common population variance based on the sample variance?

g.) Using the 0.01 level of significance, what is the critical value of the F statistic for the ANOVA test? Round your answer to atleast three decimal places.

h.) Can we conclude, using the 0.01 level of significance, that there is a difference in the mean daily sales among the different locations? YES or NO

5. We're interested in examining whether or not there are differences in the typical work week length based on the level of education of the worker. We perform a one-way, independent-samples ANOVA test, with the "groups" in the ANOVA being categories of highest educational degree of the worker ("less than high school," "bachelor's degree," etc.) and the variable being the number of hours written down by the worker in answering the survey question.

Suppose that the results of our ANOVA are given in the ANOVA table below. Complete the missing cell of the ANOVA table (rounding your answer to two decimal places), and then answer the questions below the table.

Source of Variation Degrees of Freedom Sum of Squares Mean Squares F Statistics

Treatments (Between Groups) 4 1307.9 327 ____?____

Error (Within Groups) 1097 111077.6 101.3

Total

1101 112385.5

i.) How many groups (categories of highest degree earned) of respondent were examined in the ANOVA?

j.) For the ANOVA test, it is assumed that the population variance of hours worked per week is the same for each population of workers represented. What is an unbiased estimate of this common population variance based on the sample variance?

k.) Using the 0.01 level of significance, what is the critical value of the F statistic for the ANOVA test? Round your answer to at least three decimal places.

l.) From the survey data can we conclude that at least one of the population differs significantly from the others in mean hours worked in a typical week? Use the 0.01 level of significance.

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This answers numerous questions related to ANOVA.