Exam 16: Analysis of Variance

One-way ANOVA is applied to three independent samples having means 12, 15 and 20, respectively. If each observation in the third sample were increased by 40, the value of the F-statistic would increase by 40.

In employing the randomised block design, the primary interest lies in reducing the within-treatments variation in order to make easier to detect differences between the treatment means.

Which of the following best describes the distribution of the test statistic for ANOVA?

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

The randomised block design with exactly two treatments is equivalent to a two-tailed:

A pharmaceutical manufacturer has been researching new formulas to provide quicker relief of minor pains. His laboratories have produced three different formulas, which he wanted to test. Fifteen people who complained of minor pains were recruited for an experiment. Five were given formula 1, five were given formula 2, and the last five were given formula 3. Each was asked to take the medicine and report the length of time until some relief was felt. The results are shown below. Do these data provide sufficient evidence to indicate that differences in the time of relief exist among the three formulas? Use $\alpha$ = 0.05. $\quad$$\quad$$\text { Time in minutes until reliefis felt }$ Formula 1 Formula 2 Formula 3 4 2 6 8 5 7 6 3 7 9 7 8 8 1 6

If we want to conduct a test to determine whether a population mean is greater than another population mean, we:

In recent years the irradiation of food to reduce bacteria and preserve the food longer has become more common. A company that performs this service has developed four different methods of irradiating food. To determine which is best, it conducts an experiment where different foods are irradiated and the bacteria count is measured. As part of the experiment the following foods are irradiated: beef, chicken, turkey, eggs, and milk. The results are shown below. Can the company infer at the 1% significance level that differences in the bacteria count exist among the four irradiation methods? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Bacteria count }$ Food Method 1 Method 2 Method 3 Method 4 Beef 47 53 36 68 Chicken 53 61 48 75 Turkey 68 85 55 45 Eggs 25 24 20 27 Milk 44 48 38 46

A financial analyst studied the percentage rates of return of three different types of mutual funds. Random samples of percentage rates of return for five periods were taken from each fund. The results appear in the table below. $\text { Mutual Funds Percentage Rates }$ Fund A Fund B Fund C 12 4 9 15 8 3 13 6 5 14 5 7 17 4 4 Test at the 5% significance level to determine whether the mean percentage rates for the three funds differ, assuming that the distribution is normal.

If we first arrange test units into similar groups before assigning treatments to them, the test design we should use is the randomised block design.

Given the significance level 0.025, the F-value for the numbers of degrees of freedom d.f. = (8, 10) is 3.85.

Statistics practitioners use the analysis of variance (ANOVA) technique to compare two or more populations of interval data.

A partial ANOVA table in a randomised block design is shown below. Source of Variation SS df MS F Treatments * 3 * * Blocks 1256 2 * * Error * * 67.67 Total 2922 11 Can we infer at the 5% significance level that the treatment means differ?

A federal politician wants to investigate the difference in average weekly losses of workers due to accidents between South Australia (SA) and Western Australia (WA). The following table shows the average weekly losses of worker hours due to accidents in 2012 at five randomly selected manufacturing firms in each of SA and WA. W 35 39 73 83 55 34 125 98 33 42 Assume that the weekly losses of worker hours are normally distributed. Is there a significant difference between the average weekly losses of workers hours due to accidents between SA and WA? Test at the 5% significance level. (Hint: Perform an F-test for one-way ANOVA to determine whether the population means differ.)

Two independent samples of size 30 each have been selected at random from the female and male students of a university. To test whether there is any difference in the grade point average between female and male students, an equal-variances t-test will be considered. Another test to consider is ANOVA. Which of the following is the most likely ANOVA to fit this test situation?

In order to examine the differences in age of teachers among five school districts, an educational statistician took random samples of six teachers' ages in each district. The data are listed below. $\text { Ages of teachers among five school districts }$ 1 2 3 4 5 42 23 45 31 37 22 52 48 45 40 33 32 50 33 45 32 44 45 38 61 26 39 37 29 50 24 33 49 46 50 Assume that ages of teachers are normally distributed. Use Tukey's multiple comparison method to determine which means differ.

In an ANOVA test, the test statistic is F = 3.08. The rejection region is F > 3.07 for the 5% level of significance, F > 3.82 for the 2.5% level, and F > 4.87 for the 1% level. For this test, the p-value is:

The purpose of designing a randomised block experiment is to reduce the between-treatments variation (SST) to more easily detect differences between the treatment means.

When the data are obtained through a controlled experiment in the single-factor ANOVA, we call the experimental design the completely randomised design of the analysis of variance.

In the randomised block design ANOVA, the sum of squares for error equals: