Hands-on soap bubble experiments are a great way to teach mathematics. In an effort to find the best possible bubble solution, two different soap concentrations were used along with two different amounts of glycerin additive. Students were then given a flat glass plate and a straw and were asked to blow their best bubbles. The diameters of the resulting bubbles (in millimeters) are listed below. Can an interaction be concluded between the soap solution and the glycerin? Is there a difference in the mean length of bubble diameter with respect to the concentration of soap to water? With respect to the amount of glycerin additive? Use $\alpha=0.05$.

$$\begin{array}{|l|l|} \hline &+1\text{ Unit glycerin} &+2\text{ Units glycerin}\\ \hline \text{Soap:water 13:25}&115,113,105,110 & 98,100,90,95\\ \hline \text{Soap:water 1:2}&90,102,100,98 & 99,100,102,95 \\ \hline \end{array}$$

Perform these steps. Assume that all variables are normally or approximately normally distributed, that the samples are independent and that the population variances are equal.

a. State the hypotheses. b. Find the critical value for each $F$ test. c. Complete the summary table and find the test value. d. Make the decision. e. Summarize the results. (Draw a graph of the cell means if necessary.)

How can we help wood surfaces resist weathering, especially when restoring historic wooden buildings? In a study of this question, researchers prepared wooden panels and then exposed them to the weather. Here are some of the variables recorded: type of wood (yellow poplar, pine, cedar); type of water repellent (solvent-based, water-based); paint thickness (millimeters); paint color (white, gray, light blue); weathering time (months). Identify each variable as categorical or quantitative.

A recent online poll posed the question: "Should female athletes be paid the same as men for the work they do?" In all, 13,147 ($44\%$) said "Yes;' another 15,182 ($50\%$) said "No;' and the remaining 1448 said "Don't know." (a) What is the sample size for this poll? (b) That's a much larger sample than standard sample surveys. In spite of this, we can't trust the result to give good information about any clearly defined population. Why?

A poll taken during the recession in 2008 asked 600 adults a series of questions about the state of the economy and their children's future. One question was, "Do you expect your children to have a better life than you have had, a worse life, or a life about the same as yours?" The responses showed 242 better, 310 worse, and 48 about the same. Use the sign test and $\alpha=.05$ to determine whether there is a difference between the number of adults who feel their children will have a better life compared to a worse life. What is your conclusion ?

A study was done before the recent surge in gasoline prices to compare the cost to drive 25 miles for different types of hybrid vehicles. The cost of a gallon of gas at the time of the study was approximate $\$ 2.50$. Based on the information given for different models of hybrid cars, trucks, and SUVs, is there sufficient evidence to conclude a difference in the mean cost to drive 25 miles? Use $\alpha=0.05$.

Suppose that all variables are normally distributed, that the samples are independent, that the population variances are equal, and that the samples are simple random samples, one from each of the populations. Also, for each exercise, perform the following steps.

a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results, and explain where the differences in the means are.

Use the traditional method of hypothesis testing unless otherwise specified.

When three or more means are compared, you use the ___technique.

Fractures accounted for $2.7 \%$ of all U.S. emergency room visits for a total of 454,000 visits for a recent year. A random sample of weekly ER visits is recorded for three hospitals in a large metropolitan area during the summer months. At $\alpha=0.05$, is there sufficient evidence to conclude a difference in means?

Do a complete one-way ANOVA. If the null hypothesis is rejected, use either the Scheffe or Tukey test to see if there is a significant difference in the pairs of means. Assume all assumptions are met.

The amount of sodium (in milligrams) in one serving for a random sample of three different kinds of foods is listed. At the $0.05$ level of significance, is there sufficient evidence to conclude that a difference in mean sodium amounts exists among condiments, cereals, and desserts?

Suppose that all variables are normally distributed, that the samples are independent, that the population variances are equal, and that the samples are simple random samples, one from each of the populations. Also, for each exercise, perform the following steps.

a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results, and explain where the differences in the means are.

Use the traditional method of hypothesis testing unless otherwise specified.

A manufacturer finds that in producing $x$ units per day (for $0<x<100$ ), three different kinds of cost are involved:

a. A fixed cost of $\$ 1,200$ per day in wages

b. A production cost of $\$ 1.20$ per day for each unit produced

c. An ordering cost of $\frac{100}{x^2}$ dollars per day.

Express the total cost as a function of $x$ and determine the level of production that results in minimal total cost.

If the null hypothesis is rejected in ANOVA, you can use the test to see where the difference in the means is found.

a. $z$ or $t$

b. $F$ or $\chi^2$

c. Scheffé or Tukey

d. Any of the above

Shown here are the abstract and two tables from a research study entitled "Adult Children of Alcoholics: Are They at Greater Risk for Negative Health Behaviors?" by Arlene E. Hall. Based on the abstract and the tables, answer these questions.

8. Why was the one-way ANOVA procedure used, as opposed to another test, such as the $t$ test?

Listed are the numbers of grams of carbohydrates in a random sample of eight-ounce servings of various types of juices. At the $0.01$ level of significance, is there evidence of a difference in means?

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. For these exercises, perform these steps. Assume the assumptions have been met.

a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results.

Kiplinger's listed the top 100 public colleges based on many factors. From that list, here is the average debt at graduation for various schools in four selected states. At $\alpha=0.05$, can it be concluded that the average debt at graduation differs for these four states?

The null hypothesis was rejected. Use the Scheffe test when sample sizes are unequal or the Tukey test when sample sizes are equal, to test the differences between the pairs of means. Assume all variables are normally distributed, samples are independent, and the population variances are equal.

When can the main effects of the two-way ANOVA be analyzed independently?