Analysis of Variance (57 problems)

If the null hypothesis is rejected use the Scheffe' test when the sample sizes are unequal to test the differences between the means, and use the Tukey test when the sample sizes are equal. Test at the .05 significance level.

State the hypotheses and identify the claim, find the critical value(s), compute the test value, make the decision, summarize the results, and explain where the differences in means are.

The head of the English dept. wants to determine if the average number of students taking an English course differs depending upon the time of day that the court is being offered.

If the null hypothesis is rejected use the Scheffe' test when the sample sizes are unequal to test the differences between the means, and use the Tukey test when the sample sizes are equal. Test at the .05 significance level.

State the hypotheses and identify the claim, find the critical value(s), compute the test value, make the decision, summarize the results, and explain where the differences in means are.

The head of the English dept. wants to determine if the average number of students taking an English course differs depending upon the time of day that the court is being offered.

Number of grams of fiber per serving for a random sample of 3 different kinds of foods is listed. It there sufficient evidence at the 0.05 level of significance to conclude that there is a difference in mean fiber content among breakfast cereals, fruits, and vegetable?

Numbers (in thousands) of farms per state found in 3 sections of the country are listed below. Test the claim at a=.05 that the mean number of farms is the same across these 3 geographic divisions.

Lengths (in feet) of a random sample of suspension bridges in the US, Europe, and Asia are shown. At a=.05, is there sufficient evidence to conclude that there is a difference in mean lengths?

a. Calculate the means

b. The MINITAB ANOVA printout is shown here. Test for interaction at the α = 0.05 level of significance.

Analysis of variance for response.

Psychologists have found that people are generally reluctant to transmit bad news to their peers. This phenomenon has been termed the "MUM effect." To investigate the cause of the MUM effect, undergraduates at a certain university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his/her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Five subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions. Then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) Describe the experiment, including the response variable, factors, factor levels, replications, and treatments.

The following is partial computer output from an ANOVA analysis. The research question being asked is whether there is a difference in Self-confidence among the subjects with respect to employment status. I will now provide you with some pertinent information.

Sums of Squares Between Groups = 2847.348

Total Sums of Squares = 78572.998

Total degrees of freedom (df) = 429

Significance (Sig.) = .000

Please do/answer the following:

1. Construct the ANOVA table with all values.

2. Is there a statistical difference among the three categories? (What number do you look at to determine this?)

3. Are all the assumptions of the ANOVA test met? (Where do you find this on the output?)

Fabric Flammability Tests in Different Laboratories. Flammability tests were conducted on children's sleepwear. The Vertical Semirestrained Test was used, in which pieces of fabric were burned under controlled conditions. After the burning stopped, the length of the charred portion was measured and recorded. The same fabric samples were tested at five different laboratories. The analysis of variance results from Excel are shown below.

a. What is the null hypothesis?

b. What is the alternative hypothesis?

c. Identify the value of the test statistic.

d. Find the critical value for a 0.05 significance level.

e. Identify the P-value.

f. Is there sufficient evidence to support the claim that the means for the different laboratories are not all the same?

What is the main principle behind analysis of variance?

The city has asked schools to test water in their drinking taps for levels of lead. Below is a table of lead levels based on random sampling conducted in fours schools (measured in parts per billion). An ANOVA analysis was performed on the data.

a) Use the values from the ANOVA output above to determine if the data provide sufficient evidence to indicate a difference in mean levels of lead between the schools. Show your hypotheses. You will need to determine the missing values for the ANOVA output (those things indicated by ???).

b) Compare the mean lead levels in School B and School C. Use a 95% confidence interval.

A medical researcher wishes to try three different techniques to lower the blood pressure of patients with high blood pressure. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, each subject’s blood pressure is recorded. Test the claim that there is no difference among the means using a One-Way ANOVA Test with a = 0.05.

The manager of a store wants to decide what kind of hand-knit sweaters to sell. The manager is considering three kinds of sweaters: Irish, Peruvian, and Shetland. The decision will depend on the results of an analysis of which kind of sweater, if any, lasts the longest before wearing out. The manager has some data collected from various customers who in the past bought different sweaters and reported how many years their sweaters lasted before wearing out. There are 20 observations on Irish sweaters, 18 on Peruvian sweaters, and 21 on Shetland sweaters. The data are assumed to be independent random samples from the three populations of sweaters. The manager hires a statistician, who carries out an ANOVA and finds SSE = 1,240 and SSTR = 740. Construct a complete ANOVA table, and determine whether there is evidence to conclude that the three kinds of sweaters do not have equal average durability.

Durable press cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. A study compared the breaking strength of untreated fabric with that of fabrics treated by three commercial durable press processes. Five specimens of the same fabric were assigned at random to each group. Here are the data in pounds:

We want to know if there is a significant difference in breaking strength among the given treatments. Use alpha = 0.05.

Number of grams of fiber per serving for a random sample of 3 different kinds of foods is listed. It there sufficient evidence at the 0.05 level of significance to conclude that there is a difference in mean fiber content among breakfast cereals, fruits, and vegetable?

A physician who specializes in weight control has three different diets she recommends. As an experiment, she randomly selected 15 patients and then assigned 5 to each diet. After three weeks the following weight losses, in pounds, were noted. At the .05 significance level, can she conclude that there is a difference in the mean amount of weight loss among the three diets?

The data below represent the weight losses for people on three different exercise programs. At Alpha = 0.01, does it appear that a difference exists in the true mean weight loss produced by the three exercise programs?

PROGRAM A	PROGRAM B	PROGRAM C
2.5	5.8	4.3
8.8	4.9	6.2
7.3	1.1	5.8
9.8	7.8	8.1
5.1	1.2	7.9

REV: 03/06

A student of the author lives in a home with a solar electric system. At the same time each day, she collected voltage reading from a meter connected to they system and the results are listed in the accompanying table. Use a 0.05 significance level to test the claim that the mean voltage reading is the same for the three different types of day. Is there sufficient evidence to support a claim of different population means? We might expect that a solar system would provide more electrical energy on sunny days than on cloudy or rainy days. Can we conclude that sunny days result in greater amounts of electrical energy.

If null hypothesis is rejected in this problem use Scheffe test if the sample sizes are unequal to test the differences between the means, and use the Tukey test if the sample sizes are equal. For this problem, state hypotheses and identify the claim, find the critical value(s), compute the test value, make the decision, summarize the results.

The weights in ounces of 4 types of women's shoes are shown here. At a=0.05 test the claim that there is no difference in the mean weights of the groups.

he chest deceleration data (g) are given below. Use a 0.05 significance level to test the null hypothesis that the different weight categories have the same mean. Do the data suggest that larger cars are safer?

Subcompact	55	47	59	49	42
Compact	57	57	46	54	51
Midsize	45	53	49	51	46
Full-size	44	45	39	58	44