Chapter 11: Inference for Distributions of Categorical Data

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the new school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of $100$ students

and asks them, “Which type of food do you prefer: Ramen, tacos, pizza, or hamburgers?” Here are her data:

An appropriate null hypothesis to test whether the food choices are equally popular is

a. $H_0:\mu =25$where $\mu =$the mean number of students that prefer each type of food.

b. $H_0:p=0.25$where p = the proportion of all students who prefer ramen.

c. $H_0:nR=nT=nP=nH=25$where $nR$is the number of students in the school who would choose ramen, and so on.

d.$H_0:pR=pT=pP=pH=0.25$where $pR$is the proportion of students in the school who would choose ramen, and so on.

e. $H_0:p^R=p^T=p^P=p^H=0.25$, where $p^R$is the proportion of students in the sample who chose ramen, and so on.

Roulette Casinos are required to verify that their games operate as advertised. American roulette wheels have $38$ slots—$18$ red, $18$ black, and $2$ green. In one casino, managers record data from a random sample of $200$ spins of one of their American roulette wheels. The table displays the results.

a. State appropriate hypotheses for testing whether these data give convincing evidence that the distribution of outcomes on this wheel is not what it should be.

b. Calculate the expected count for each color.

c. Calculate the value of the chi-square test statistic.

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the new school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of $100$students and asks them, “Which type of food do you prefer: Ramen, tacos, pizza, or hamburgers?” Here are her data:

The chi-square test statistic is

a. $\frac{{(18-25)}^2}{25}+\frac{{(22-25)}^2}{25}+\frac{{(39-25)}^2}{25}+\frac{{(21-25)}^2}{25}$

b. $\frac{{(25-18)}^2}{18}+\frac{{(25-22)}^2}{22}+\frac{{(25-39)}^2}{39}+\frac{{(25-21)}^2}{21}$

c. $\frac{(18-25)}{25}+\frac{(22-25)}{25}+\frac{(39-25)}{25}+\frac{(21-25)}{25}$

d. $\frac{{(18-25)}^2}{100}+\frac{{(22-25)}^2}{100}+\frac{{(39-25)}^2}{100}+\frac{{(21-25)}^2}{100}$

e. $\frac{{(0.18-0.25)}^2}{0.25}+\frac{{(0.22-0.25)}^2}{0.25}+\frac{{(0.39-0.25)}^2}{0.25}+\frac{{(0.21-0.25)}^2}{0.25}$

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the new school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of $100$students and asks them, “Which type of food do you prefer: Ramen, tacos, pizza, or hamburgers?” Here are her data:

The P-value for a chi-square test for goodness of fit is 0.0129. Which of the following is the most appropriate conclusion at a significance level of 0.05?

a. Because $0.0129$ is less than $\alpha =0.05$ reject $H_0$ . There is convincing evidence that the food choices are equally popular.

b. Because $0.0129$ is less than $\alpha =0.05$ reject $H_0$ There is not convincing

evidence that the food choices are equally popular.

c. Because $0.0129$ is less than $\alpha =0.05$ reject $H_0$ . There is convincing evidence that the food choices are not equally popular.

d. Because $0.0129$ is less than $\alpha =0.05$ fail to reject $H_0$ There is not convincing evidence that the food choices are equally popular.

e. Because $0.0129$ is less than $\alpha =0.05$ fail to reject $H_0$ There is convincing

evidence that the food choices are equally popular.

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the new school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of $100$students and asks them, “Which type of food do you prefer: Ramen, tacos, pizza, or hamburgers?” Here are her data:

Which of the following is false?

a. A chi-square distribution with k degrees of freedom is more right-skewed than a chi square distribution with $k+1$ degrees of freedom.

b. A chi-square distribution never takes negative values.

c. The degrees of freedom for a chi-square test are determined by the sample size.

d. $P(\chi 2>10)$ is greater when $df=k+1$ than when $df=k$

e. The area under a chi-square density curve is always equal to $1$

Video games $(1.1)$ To determine if there is a relationship between age and playing video games, the Pew Research Center asked randomly selected adults for their age and if they “ever play video games on a computer, TV, game console, or portable device like a cell phone.”$3$ Here is a two-way table summarizing the results of the study:

a. Construct a segmented bar graph to display the relationship between age group and response to the question about video games.

b. Describe the association shown in the segmented bar graph in part (a).

Do students who read more books for pleasure tend to earn higher grades in English? The boxplots show data from a simple random sample of $79$ students at a large high school. Students were classified as light readers if they read fewer than $3$ books for pleasure per year. Otherwise, they were classified as heavy readers. Each student’s average English grade for the previous two

marking periods was converted to a GPA scale, where $A=4.0,A-=3.7,B+=3.3$

and so on.

Do students who read more books for pleasure tend to earn higher grades in English? The boxplots show data from a simple random sample of $79$ students at a large high school. Students were classified as light readers if they read fewer than $3$ books for pleasure per year. Otherwise, they were classified as heavy readers. Each student’s average English grade for the previous two

marking periods was converted to a GPA scale, where$A=4.0,A-=3.7,B+=3.3$

Reading and grades (10.2) Summary statistics for the two groups from Minitab are provided.

a. Explain why it is acceptable to use two-sample $t$ procedures in this setting.

b. Construct and interpret a $95\%$ confidence interval for the difference in the mean English grade for light and heavy readers.

c. Does the interval in part (b) provide convincing evidence that reading more causes a difference in students’ English grades? Justify your answer.

Do students who read more books for pleasure tend to earn higher grades in English? The boxplots show data from a simple random sample of 79 students at a large high school. Students were classified as light readers if they read fewer than 3 books for pleasure per year. Otherwise, they were classified as heavy readers. Each student’s average English grade for the previous two

marking periods was converted to a GPA scale, where A=4.0, A−=3.7, B+=3.3

Reading and grades (3.2) The scatterplot shows the number of books read and the English grade for all 79 students in the study. The least-squares regression line y^=3.42+0.024x has been added to the graph.

a. Interpret the slope and y intercept.

b. The student who reported reading 17 books for pleasure had an English GPA of 2.85. Calculate and interpret this student’s residual.

c. For this linear model, r2=0.083 Interpret this value.

The color of candy Inspired by the example about how background music influences

choice of entrée at a restaurant, a statistics student decided to investigate other ways to

influence a person’s behavior. Using $60$volunteers, she randomly assigned $20$volunteers

to get a “red” survey, $20$volunteers to get a “blue” survey, and $20$volunteers to get a

control survey. The first three questions on each survey were the same, but the fourth and

fifth questions were different. For example, the fourth question on the “red” survey was

“When you think of the color red, what do you think about?” On the blue survey, the

question replaced red with blue. On the control survey, the last two questions were not

about color. As a reward, each volunteer was allowed to choose a chocolate candy in a red

wrapper or a chocolate candy in a blue wrapper. Here are segmented bar graphs showing

the results of the experiment. Describe what you see.