Exam 16: Analysis of Variance

In recent years a controversy has arisen in major league baseball in the US. Some players have been accused of 'doctoring' their bats to increase the distance the ball travels. However, a physics professor claims that the effect of doctoring is negligible. A major league manager decides to test the professor's claim. He doctors two bats by inserting cork into one and rubber into another. He then tells five players on his team to hit a ball with an undoctored bat and with the doctored bats. The distances are measured, and are listed below. Do these data provide sufficient evidence at the 5% level of significance to refute the professor's claim? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Distance ball travels (feet) }$ Player Undoctored bat Bat with cork Bat with rubber 1 275 265 280 2 315 335 320 3 425 435 440 4 380 375 370 5 450 460 450

Three tennis players, one a beginner, one intermediate and one advanced, have been randomly selected from the membership of a club in a large city. Using the same tennis ball, each player hits ten serves, one with each of three racquet models, with the three racquet models selected randomly. The speed of each serve is measured with a machine and the result recorded. Among the ANOVA models listed below, the most likely model to fit this situation is the:

Which of the following statements is true?

A statistician employed by a television rating service wanted to determine whether there were differences in television viewing habits among three residential areas in South Australia. He took a random sample of five adults in each of the areas and asked each to report the number of hours spent watching television in the previous week. From the data shown below, can he infer at the 5% level of significance that differences in hours of television watching exist among the three residential areas? Henley Beach West Lakes Shore Glenelg 25 30 23 25 33 18 18 40 21 15 29 20 27 36 10

Provide an example of a randomised block design with three treatments (k = 3) and four blocks (b = 4), in which SSB = 0 and SST and SSE are not equal to zero.

The test statistic of the single-factor ANOVA equals:

Which of the following best describes the between-treatments in single-factor analysis of variance?

A study is to be undertaken to examine the effects of two kinds of background music and of two assembly methods on the output of workers at a fitness shoe factory. Two workers will be randomly assigned to each of four groups, for a total of eight in the study. Each worker will be given a headphone set so that the music type can be controlled. The number of shoes completed by each worker will be recorded. Does the kind of music or the assembly method or a combination of music and method affect output? The ANOVA model most likely to fit this situation is the two-way analysis of variance.

In ANOVA, the between-treatments variation is denoted by SST, which stands for sum of squares of treatments.

The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample sizes are equal.

The F-statistic in a one-way ANOVA represents the variation between the treatments divided by the variation within the treatments.

The marketing management of a shopping complex food court, wants to investigate the average age of customers of three of the food court's fast food outlets: fish and chips, sandwich bar, and chicken and chips. This is to ascertain whether they should be providing different seating options for patrons. They take a random sample of eight customers at each of these three outlets and record their ages. Do these data provide enough evidence at the 5% significance level to infer that there are differences in ages among the customers of the fast food outlets, and that the food court should provide different seating options? From previous analyses, ages of customers of this food court are normally distributed.

Provide an example for a randomised block design with three treatments (k = 3) and four blocks (b = 4), in which SST is equal to zero and SSB and SSE are not equal to zero.

In one-way ANOVA, suppose that there are five treatments with $n _ { 1 } = n _ { 2 } = n _ { 3 } = 5$ and $n _ { 4 } = n _ { 5 } = 7$ . Then the mean square for error, MSE, equals:

A random sample of 10 observations was selected from each of four normal populations. A partial one-way ANOVA table is shown below: Source of Variation SS df MS F Treatments * * 270 * Error * * * Total 1,350 * a. Complete the missing entries (identified by asterisks) in the ANOVA table. b. How many groups were in this study? c. How many experimental units were in this study? d. At the 5% significance level, can we infer that the means of the populations differ?