Gastric freezing was once a recommended treatment for ulcers in the upper intestine. Use of gastric freezing stopped after experiments showed it had no effect. One randomized comparative experiment found that 28 of the 82 gastric-freezing patients improved, while 30 of the 78 patients in the placebo group improved. We can test the hypothesis of "no difference" in the effectiveness of the treatments in two ways: with a two-sample z test or with a chi-square test. State appropriate hypotheses for a chi-square test.

Gastric freezing was once a recommended treatment for ulcers in the upper intestine. Use of gastric freezing stopped after experiments showed it had no effect. One randomized comparative experiment found that 28 of the 82 gastric-freezing patients improved, while 30 of the 78 patients in the placebo group improved. We can test the hypothesis of "no difference" in the effectiveness of the treatments in two ways: with a two-sample z test or with a chi-square test. Here is Minitab output for a chi-square test. Interpret the P-value. What conclusion would you draw? Chi-Square Test: Generic freezing, placebo. Expected counts are printed below observed counts. Chi-Square contributions are printed below expected counts.

$$\begin{array}{l} \hline \text{ } & \text{Gastric}\\ \text{ } & \text{freezing} & \text{placebo} & \text{Total}\\ \text{Improved} & \text{28} & \text{30} & \text{58}\\ \text{ } & \text{29.73} & \text{28.27}\\ \text{ } & \text{0.1} & \text{0.105}\\ \text{Didn't improve} & \text{54} & \text{48} & \text{102}\\ \text{ } & \text{52.27} & \text{49.73}\\ \text{ } & \text{0.057} & \text{0.06}\\ \text{Total} & \text{82} & \text{78} & \text{160}\\ \text{Chi-Sq =0.322} & \text{DF=1} & \text{P-Value =0.570}\\ \hline \end{array}$$

Use your calculator's RandInt function to generate 200 digits from 0 to 9 and store them in a list. State appropriate hypotheses for a chi-square test for goodness of fit to determine whether your calculator's random number generator gives each digit an equal chance to be generated.

Justin wondered if the boys and girls in his school have different preferences for animated shows on a popular television network. He asked a simple random sample of 75 students at his school to choose their favorite show from a list of four shows, recording their gender as well. Here are his data. Check the conditions for performing a chi-square test for association.

$$\begin{array} {lc} & \text{Gender} \\ \text {$\begin{matrix} \text{Preferred}\\ \text{show}\\ \end{matrix}$} & \begin{array}{l|c|c|c} & \text { Male } & \text { Female } & \text { Total } \\ \hline^{\prime \prime} \mathrm{RR}^{\prime \prime} & 8 & 10 & 18 \\ \hline^{\prime \prime} \mathrm{RP}^{\prime \prime} & 10 & 13 & 23 \\ \hline^{\prime \prime} \mathrm{HA}^{\prime \prime} & 5 & 5 & 10 \\ \hline^{\prime \prime} \mathrm{AB}^{\prime \prime} & 7 & 5 & 12 \\ \hline \text { No opinion } & 5 & 7 & 12 \\ \hline \text { Total } & 35 & 40 & 75 \end{array}\\ \end{array}$$

A software company develops and markets a popular business simulation/modeling program. A random number generator contained in the program provides random values from various probability distributions. The software design group would like to validate that the program is properly generating random numbers. Accordingly, they generated 5,000 random numbers from a normal distribution and grouped the results into the accompanying frequency distribution. The sample mean and sample standard deviation are 100 and 10 ,

$$\begin{array}{|l|c|} \hline \text { Value } & \text { Frequency } \\ \hline \text { Under } 70 & 12 \\ \hline 70 \text { up to } 80 & 99 \\ \hline 80 \text { up to } 90 & 658 \\ \hline 90 \text { up to } 100 & 1734 \\ \hline 100 \text { up to } 110 & 1681 \\ \hline 110 \text { up to } 120 & 697 \\ \hline 120 \text { up to } 130 & 112 \\ \hline 130 \text { or more } & 7 \\ \hline & \text { Total }=5,000 \\ \hline \end{array}$$

a. Using the goodness-of-fit test for normality, state the competing hypotheses to test if the random numbers generated do not follow the normal distribution.

b. Calculate the value of the test statistic and the $p$-value.

c. What is the conclusion to the test at the $1 \%$ significance level? Is your conclusion sensitive within the range of typical significance levels?

A broth used to manufacture a pharmaceutical product has its sugar content, in mg/mL, measured several times on each of three successive days.

$$\begin{matrix} \text{Day 1:} & \text{5.0} & \text{4.8} & \text{5.1} & \text{5.1} & \text{4.8} & \text{5.1} & \text{4.8}\\ \quad & \text{4.8} & \text{5.0} & \text{5.2} & \text{4.9} & \text{4.9} & \text{5.0}\\ \text{Day 2:} & \text{5.8} & \text{4.7} & \text{4.7} & \text{4.9} & \text{5.1} & \text{4.9} & \text{5.4}\\ \quad & \text{5.3} & \text{5.3} & \text{4.8} & \text{5.7} & \text{5.1} & \text{5.7}\\ \text{Day 3:} & \text{6.3} & \text{4.7} & \text{5.1} & \text{5.9} & \text{5.1} & \text{5.9} & \text{4.7}\\ & \text{6.0} & \text{5.3} & \text{4.9} & \text{5.7} & \text{5.3} & \text{5.6}\\ \end{matrix}$$

a. Can you conclude that the variability of the process is greater on the second day than on the first day? b. Can you conclude that the variability of the process is greater on the third day than on the second day?

A survey of 4826 randomly selected young adults (aged 19 to 25 ) asked, "What do you think the chances are you will have much more than a middle-class income at age 30? The table shows the responses. State the hypotheses for a test of the relationship between gender and opinion about getting rich for young adults.

$$\begin{array} {lc} & \text{Gender} \\ \text {Opinion} & \begin{array}{l|c|r|r} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \begin{array}{l} \text { Some chance but } \\ \text { probably not } \end{array} & 426 & 286 & 712 \\ \hline \text { A } 50-50 \text { chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}\\ \end{array}$$

A study of first-year college students asked separate random samples of students from private and public universities the following question "How many miles is this university from your permanent house?" Students had to choose from the following options: 5 or fewer, 6 to 10, 11 to 50, 51 to 100, 101 to 500, or more than 500. Here is the two-way table summarizing the responses.

$$\begin{array}{ccc} & \text{Type of university} \\ \text{Distạnce from home (miles)} & {\begin{array}{l|r|r|r} & \text { Public } & \text { Private } & \text { Total } \\ \hline 5 \text { or fewer } & 1951 & 1028 & 2979 \\ \hline 6 \text { to } 10 & 2688 & 1285 & 3973 \\ \hline 11 \text { to } 50 & 10,971 & 5527 & 16,498 \\ \hline 51 \text { to } 100 & 6765 & 2211 & 8976 \\ \hline 101 \text { to } 500 & 15,177 & 6195 & 21,372 \\ \hline \text { Over 500 } & 5811 & 9486 & 15,297 \\ \hline \text { Total } & 43,363 & 25,732 & 69,095 \end{array}} \\ \end{array}$$

Should we use a chi-square test for homogeneity or chi-square test for independence in this settings? Justify your answer.

A random sample of traffic tickets given to motorists in a large city is examined. The tickets are classified according to the race or ethnicity of the driver. The results are summarized in the following table.

$$\begin{array}{lcccc} \hline \text { Race/Ethnicity } & \text { White } & \text { Black } & \text { Hispanic } & \text { Other } \\ \text { Number of tickets } & 69 & 52 & 18 & 9 \\ \hline \end{array}$$

The proportion of this city's population in each of the racial/ethnic categories listed is as follows.

$$\begin{array}{lcccc} \hline \text { Race/Ethnicity } & \text { White } & \text { Black } & \text { Hispanic } & \text { Other } \\ \text { Proportion } & 0.55 & 0.30 & 0.08 & 0.07 \\ \hline \end{array}$$

We wish to test $H_{0}:$ The racial/ethnic distribution of traffic tickets in the city is the same as the racial/ethnic distribution of the city's population.

Who were the popular kids at your elementary school? Did they get good grades or have good looks? Were they good at sports? A study was performed in Michigan to examine the factors that determine social status for children in grades 4, 5, and 6. Researchers administered a questionnaire to a random sample of 478 students in these grades. One of the questions asked, "What would you most like to do at school make good grades, be good at sports, or be popular?" The two-way table summarizes the students' responses. ls there convincing evidence of an association between gender and goal for students in grades 4, 5, and 6?

$$\text { Gender }\\ \text { Goal } \begin{array}{l|c|c|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Grades } & 130 & 117 & 247 \\ \hline \text { Popular } & 91 & 50 & 141 \\ \hline \text { Sports } & 30 & 60 & 90 \\ \hline \text { Total } & 251 & 227 & 478 \end{array}$$

Choose the best answer. After a name-brand drug has been sold for several years, the Food and Drug Administration (FDA) will allow other companies to produce a generic equivalent. The FDA will permit the generic drug to be sold as long as there isn't convincing evidence that it is less effective than the name brand drug. For a proposed generic drug intended to lower blood pressure, the following hypotheses will be used:

$$H_0: \mu_G = \mu_N$$

versus

$$H_a: \mu_G < \mu_N$$

where

$$\mu_G =$$

true mean reduction in blood pressure using the generic drug,

$$\mu_N =$$

true mean reduction in blood pressure using the name-brand drug. In the context of this situation, which of the following describes a Type I error? (a) The FDA finds convincing evidence that the generic drug is less effective, when in reality it is less effective. (b) The FDA finds convincing evidence that the generic drug is less effective, when in reality it is equally effective. (c) The FDA fails to find convincing evidence that the generic drug is less effective, when in reality it is less effective. (d) The FDA fails to find convincing evidence that the generic drug is less effective, when in reality it is equally effective. (e) The FDA finds convincing evidence that the generic drug is equally effective, when in reality it is less effective.

Sally's Software Inc. is a rapidly growing supplier of computer software to the Sarasota area. Sales for the last 5 years, 2013 to 2017, are given below.

a. Determine the logarithmic trend equation.

b. By what percent did sales increase, on the average, during the period?

c. Estimate sales for the year 2020.

An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table.

$$\begin{matrix} \text{ } & \text{ } & \text{ } & \text{y} & \text{ } & \text{ }\\ \text{p(x, y)} & \text{ } & \text{0} & \text{5} & \text{10} & \text{15}\\ \text{ } & \text{0} & \text{.02} & \text{.06} & \text{.02} & \text{.10}\\ \text{x} & \text{5} & \text{.04} & \text{.15} & \text{.20} & \text{.10}\\ \text{ } & \text{10} & \text{.01} & \text{.15} & \text{.14} & \text{.01}\\ \end{matrix}$$

a. If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score E(X + Y)? b. If the maximum of the two scores is recorded, what is the expected recorded score?

You read a newspaper article that describes a study of whether stress management can help reduce heart attacks. The 107 subjects all had reduced blood flow to the heart and so were at risk of a heart attack. They were assigned at random to three groups. The article goes on to say: One group took a four-month stress management program, another underwent a four-month exercise program, and the third received usual heart care from their personal physicians. In the next three years, only 3 of the 33 people in the stress management group suffered "cardiac events," defined as a fatal or non-fatal heart attack or a surgical procedure such as a bypass or angioplasty. In the same period, 7 of the 34 people in the exercise group and 12 out of the 40 patients in usual care suffered such events. Do the data provide convincing evidence at the $\alpha=0.05$ level that the true success rates for patients like these are not the same for the three treatments?