# ANOVA

1. A manufacturer of top-of the line tennis rackets claims that its Smack Me tennis racket will change a player's game. Tennis pro currently serves the ball at an average speed at 115 mph with a standard deviation of 2.5 mph. The speeds are normally distributed. The tennis pro decides to test the company's claim and records the speed of his serve for 15 balls using the Smack Me racket. The data are shown in the following table:

Speed (mph)

117.3 115.9

115.1 115.2

116.0 115.0

116.2 113.0

112.9 120.8

115.4 116.9

113.8 114.4

114.2

2. What is the theory underlying ANOVA?

3. The mean length of a small counterbalance bar is 43 millimeters. The production supervisor is concerned that the adjustments of the m machine producing the bars have changed. He asks the Engineering Department to investigate. Engineering selects a random sample of 12 bars and measure each. The results are reported below in millimeters. 42 39 42 45 43 40 39 41 40 42 43 42

Is it reasonable to conclude that there has been a change in the mean length of the bars? Use the .02 significance level.

4. Why do we say that ANOVA compares samples' means, when we are actually comparing variances?

5. In ANOVA, what does F=1 mean?

© BrainMass Inc. brainmass.com October 24, 2018, 5:32 pm ad1c9bdddfhttps://brainmass.com/statistics/analysis-of-variance/13021

#### Solution Summary

1. A manufacturer of top-of the line tennis rackets claims that its Smack Me tennis racket will change a player's game. Tennis pro currently serves the ball at an average speed at 115 mph with a standard deviation of 2.5 mph. The speeds are normally distributed. The tennis pro decides to test the company's claim and records the speed of his serve for 15 balls using the Smack Me racket. The data are shown in the following table:

Speed (mph)

117.3 115.9

115.1 115.2

116.0 115.0

116.2 113.0

112.9 120.8

115.4 116.9

113.8 114.4

114.2

2. What is the theory underlying ANOVA?

3. The mean length of a small counterbalance bar is 43 millimeters. The production supervisor is concerned that the adjustments of the m machine producing the bars have changed. He asks the Engineering Department to investigate. Engineering selects a random sample of 12 bars and measure each. The results are reported below in millimeters. 42 39 42 45 43 40 39 41 40 42 43 42

Is it reasonable to conclude that there has been a change in the mean length of the bars? Use the .02 significance level.

4. Why do we say that ANOVA compares samples' means, when we are actually comparing variances?

5. In ANOVA, what does F=1 mean?

ANOVA Problem

When we want to test two samples to determine if it is likely that the population means (estimated by the sample means) are different, we typically use a t-test. If the samples are large, we can also use a z-test. (Note that the formulas for computing s, t and/or z in the case of a two-sample test are different than the formulas for computing the same values in a one-sample test. Use Excel data analysis to conduct tests comparing two sample means.)

Using ANOVA (short for Analysis of Variance), however, we can test 3 or more sample means to determine if at least one of the sample means comes from a population with a mean that is significantly different from all of the others in the test. We actually do this by estimating a combined population variance two different ways and comparing the two estimates (the ratio of these two variance estimates follows the so-called "F distribution").

Question:

Why do we need a new test method to compare the means of 3 or more populations? Why can't we just use a series of z-tests or t-tests to compare all of the possible pairs of population means to see if one (or more) is different?

Most of the testing is to determine one or two things:

1. Is there a statistically significant difference between two or more population means? (based on comparison of 2 or more sample means)

2. Is there a statistically significant relationship between two or more variables? We can use regression analysis or chi-square tests to answer this second question.)

View Full Posting Details