Chapter 12 Section 1
Homework Set A
12.9 Describing the ANOVA model. For each of the following situations, identify the response variable and the populations to be compared, and give I, the ni, and N.
(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 25 of his hens to each of the three treatments.
(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a five-point scale. She received 94 responses, of which 31 were from students who attend varsity football or basketball games only, 18 were from students who also attend other varsity competitions, and 45 who did not attend any varsity games.
(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 42 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student a quiz on power calculations, and these scores were compared.
12.11 Determining the degrees of freedom. Refer to Exercise 12.9. For each situation, give the following:
(a) Degrees of freedom for the model, for error, and for the total.
(b) Null and alternative hypotheses.
(c) Numerator and denominator degrees of freedom for the F statistic.
12.15 A one-way ANOVA example. A study compared 4 groups with 8 observations per group. An F statistic of 3.33 was reported.
(a) Give the degrees of freedom for this statistic and the entries from Table E (optional. Do this if you think if will help your understanding) that corresponds to this distribution.
(b) Sketch a picture of this F distribution using the applet at
http://www.stat.tamu.edu/~west/applets/fdemo.html
Make sure you label the horizontal axis, and the vertical axis.
(c) Based on the table information, how would you report the P- value?
(d) Can you conclude that all pairs of means are different? Explain your answer.
12.17 Calculating the ANOVA F test P-value, continued. For each of the following situations, find the F statistic and the degrees of freedom. Then draw a sketch of the distribution (use the applet found at
http://www.stat.tamu.edu/~west/applets/fdemo.html ) under the null hypothesis and shade in the portion corresponding to the P-value. State how you would report the P-value.
(use the table I made in class which can also be found at page 654.)
(a) Compare 5 groups with 9 observations per group, MSE = 50, and MSG = 127.
(b) Compare 4 groups with 7 observations per group, SSG= 40, and SSE = 153.
FOR PROBLEMS 18 AND 19 YOU WILL NEED TO USE THE APPLET FOUND AT
WWW.WHFREEMAN.COM/IPS6E LOOK AT THE APPLETS SECTION, SCROLL DOWN TO THE BOTTOM; IT IS THE LAST APPLET.
12.18 The effect of increased variation within groups. The One-Way ANOVA applet lets you see how the F statistic and the P-value depend on the variability of the data within groups and the differences among the means.
(a) The black dots are at the means of the three groups. Move these up and down until you get a configuration that gives a P-value of about 0.01. What is the value of the F statistic?
(b) Now increase the variation within the groups by dragging the mark on the pooled standard error scale to the right. Describe what happens to the F statistic and the P-value. Explain why this happens.
12.19 The effect of increased variation between groups. Set the pooled standard error for the One-Way ANOVA applet at a middle value. Drag the black dots so that they are approximately equal.
(a) What is the F statistic? Give its P-value.
(b) Drag the mean of the second group up and the mean of the third group down. Describe the effect on the F statistic and its P-value. Explain why they change in this way.
12.20 Calculating the pooled standard deviation.
An experiment was run to compare four groups. The sample sizes were 25, 28, 150, and 21, and
the corresponding estimated standard deviations were 42, 38, 20, and 45.
(a) Is it reasonable to use the assumption of equal standard deviations when we analyze these data? Give a reason for your answer.
(b) Give the values of the variances for the four groups.
(c) Find the pooled variance.
(d) What is the value of the pooled standard deviation?
(e) Explain why your answer in part (d) is much closer to the standard deviation for the third group than to any of the other standard deviations.
12.34 Air quality in poultry-processing plants. The air in poultry-processing plants often contains fungus spores. If the ventilation is inadequate, this can affect the health of the workers. To measure the presence of spores, air samples are pumped to an agar plate, and "colony-forming units (CPUs)" are counted after an incubation period. Here are data from the "kill room" of a plant that slaughters 37,000 turkeys per day, taken at four seasons of the year. The units are CFUs per cubic meter of air."
Fall / Winter / Spring / Summer1231 / 987 / 2054 / 1452
1254 / 778 / 2092 / 1521
1088 / 852 / 1902 / 1352
(a) Examine the data using graphs and descriptive measures. How do airborne fungus spores vary with the seasons? As we did in class.
(b) Is the effect of season statistically significant?