Exam 12: Analysis of Variance Tests

Analysis of variance (ANOVA) is a set of techniques that allow us to compare two or more sample means at the same time.

Lawn Fertilizer To test the effectiveness of three different types of lawn fertilizer,each type was applied to six different lawns.The same size patch of lawn was mowed for all 18 test units and the grass clippings were weighed.The weight in pounds are shown in the following table. Fertilizer 1 Fertilizer 2 Fertilizer 3 Lawn 1 10.3 11.7 Lann 2 12.0 Lawn 3 9.3 10.7 Lawn 4 10.6 10.5 9.3 Lawn 5 9.1 9.8 10.4 Lawn 6 12.4 9.6 -Develop a randomized block design to test for the difference between the average number of customers per day between four different stores using $\alpha$ = 0.05 and using the days as the blocking variable. Calculate the treatment sum of squares and the treatment mean square.

Model A consumer group is testing the gas mileage of 3 different models of cars.Several cars were driven 1000 miles and the gas mileage was recorded in the following table. Madel 1 Model 2 Madel 3 22.6 19.6 25.3 20.9 22.3 21.6 23.6 20.5 20.5 21.9 17.0 24.6 22.0 17.6 22.7 21.1 19.2 20.5 18.7 20.9 -Calculate the treatment mean square and the error mean square. Calculate the test statistic.

Alumni clubs Three alumni clubs for a university have members who contribute to the university athletic program.The amount of contribution (in dollars)for six contributors randomly selected from each city to test the hypothesis that the mean contribution is the same among alumni clubs is given below,followed by the total for each city: Banks Business Services Construction =7 =7 =7 =41 =30 =24 =882 =1266 =582 -State the null and alternative hypotheses. H₀: ____________________ H₁: ______________________________________________

Three racquetball players,a beginner,an experienced,a professional,have been randomly selected from the membership of a large city racquetball club.Using the same ball,each person hits three serves,one with each of five racquet models,with the five racquet models selected randomly.Each serve is clocked with a radar gun and the result recorded.Among ANOVA models,this setup is most like the two-way ANOVA.

Stores A vice president would like to know whether there is a difference between the average number of customers per day between four different stores.The following data was collected over the same 5-day period for each store. Stare 1 Stare 2 Stare 3 Stare 4 Day 1 36 12 25 30 Day 2 42 22 51 36 Day 3 39 33 40 28 Day 4 25 30 42 46 Day 5 47 34 26 35 -Develop a randomized block design to test for the difference between the average number of customers per day between four different stores using $\alpha$ = 0.05 and using the days as the blocking variable. Calculate the block sum of squares.

Two samples of ten each from the male and female workers of a large company have been taken.The data involved the wage rate of each worker.To test whether there is any difference in the average wage rate between male and female workers a pooled-variances t-test will be considered.Another test option to consider is ANOVA.The most likely ANOVA to fit this test situation is the:

Match each term with its definition: -Mean square for the between-group variation

The F-test of the randomized block design of the analysis of variance requires that the random variable of interest must be normally distributed and the population variances must be equal.When the random variable is not normally distributed,we can use:

In employing the randomized block design,the primary interest lies in:

What is the purpose of the randomized block design?

In one-way ANOVA,suppose that there are five treatments with n₁ = n₂ = n₃ = 5,and n₄ = n₅ = 7.Then the mean square for error,MSE,equals:

Courses A father and his two sons played golf at four different courses with the following scores: Dad Sam Jeff Course 1 92 86 81 Course2 95 86 87 Course 3 88 81 83 Cuurse4 90 87 90 -Use a randomized block design to test for the difference of golf score means using $\alpha$ = 0.05 and using the courses as the blocking variable. Calculate the test statistic.

Grocery store Management of the refrigerator space in a grocery store requires knowledge of the average amount of items sold.Specifically the store manager wishes to determine whether or not to allocate the same shelf space for three flavors of yogurt: raspberry,peach,and lemon.During five randomly selected weeks the sales data for each flavor were recorded.The average daily sales for each flavor were: Week Raspberry Peach Lemon 1 39 35 46 2 26 20 42 3 33 34 19 4 23 39 25 5 27 45 36 Total 148 173 168 -Calculate the error sum of squares (SSE).

When there are only two treatments,the one-way ANOVA is equivalent to the nondirectional ______________________________ t-test.

Grocery store Management of the refrigerator space in a grocery store requires knowledge of the average amount of items sold.Specifically the store manager wishes to determine whether or not to allocate the same shelf space for three flavors of yogurt: raspberry,peach,and lemon.During five randomly selected weeks the sales data for each flavor were recorded.The average daily sales for each flavor were: Week Raspberry Peach Lemon 1 39 35 46 2 26 20 42 3 33 34 19 4 23 39 25 5 27 45 36 Total 148 173 168 -Compute the appropriate table value to use as the critical value at the 5% significance level.

Match each term with its definition: -Mean square for within-group variation

Grocery store Management of the refrigerator space in a grocery store requires knowledge of the average amount of items sold.Specifically the store manager wishes to determine whether or not to allocate the same shelf space for three flavors of yogurt: raspberry,peach,and lemon.During five randomly selected weeks the sales data for each flavor were recorded.The average daily sales for each flavor were: Week Raspberry Peach Lemon 1 39 35 46 2 26 20 42 3 33 34 19 4 23 39 25 5 27 45 36 Total 148 173 168 -Calculate the total sum of squares (SST).

Shelf Level Shelf height may affect the volume of sales for a product.In order to determine whether or not there is any effect on average sales due to product shelf height,a marketing consultant chooses four typical outlets,a supermarket,a drug store,a discount store,and a variety store,and for one week places the product chosen at one eye level.A resting week follows and the product is moved to another height and so on for three different heights.The number of units sold per week at each location and height are: Shelf Level Store Above Even Below Discount 54 55 49 Supermarket 63 60 56 Drug Store 52 52 47 Variety 44 48 41 -Compute the sum of squares for treatments (SSTR).

In two-way ANOVA,it is not appropriate to refer to the factor levels as treatments,since each combination of their levels constitutes a separate treatment.