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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 11: Q 22. (page 718)

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. $P (χ 2 > 10)$ is greater when $d f = k + 1$ than when $d f = k$
e. The area under a chi-square density curve is always equal to $1$

Short Answer

Expert verified

The correct option is (c).

Step by step solution

01

Given information

02

Calculation

Option c is wrong among the possibilities presented. The number of categories determines the degrees of freedom of a chi-square test. $d f = c - 1$ is the formula. As a result, sample size has no bearing on degrees of freedom.

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Most popular questions from this chapter

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.

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

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.

We compute the value of the $χ^{2}$ test statistic to be $6.57$ . Assuming that the conditions for inference are met, which of the following is the correct P-value?

a. Greater than $0.20$

b. Between $0.10$ and $0.20$

c. Between $0.05$ and $0.10$

d. Between $0.01$ and $0.05$

e. Less than $0.01$

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.

Relaxing in the sauna Researchers followed a random sample of $2315$ middle-aged men from eastern Finland for up to $30$ years. They recorded how often each man went to a sauna and whether or not he suffered sudden cardiac death (SCD). The two-way table shows the data from the study.

a. State appropriate hypotheses for performing a chi-square test for independence in this setting.

b. Compute the expected counts assuming that $H_{0}$ is true.

c. Calculate the chi-square test statistic, df, and P-value.

d. What conclusion would you draw?

Distance from home 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 home?” 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:

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

b. State appropriate hypotheses for performing the type of test you chose in part (a). Here is Minitab output from a chi-square test.

c. Check that the conditions for carrying out the test are met.

d. Interpret the P-value. What conclusion would you draw?

“Will changing the rating scale on a survey affect how people answer the question?” To find out, the group took an SRS of $50$ students from an alphabetical roster of the school’s just over $1000$ students. The first $22$ students chosen were asked to rate the cafeteria food on a scale of $1$ (terrible) to $5$ (excellent). The remaining $28$ students were asked to rate the cafeteria food on a scale of $0$ (terrible) to $4$ (excellent). Here are the data:

The students decided to compare the average ratings of the cafeteria food on the two scales.

a. Find the mean and standard deviation of the ratings for the students who were given the $1 t o 5$ scale.

b. For the students who were given the $0 t o 4$ scale, the ratings have a mean of $3.21$ and a standard deviation of $0.568$ . Since the scales differ by one point, the group decided to add $1$ to each of these ratings. What are the mean and standard deviation of the adjusted ratings?

c. Would it be appropriate to compare the means from parts (a) and (b) using a two-sample t test? Justify your answer

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