11 Categorical Data Analysis

Section 11.1 Exercises

CLARIFYING THE CONCEPTS

Question 11.1

1. What are the conditions required for a random variable to be multinomial? (p. 632)

11.1.1

(1) Each independent trial of the experiment has $k$ possible outcomes, $k = 2, 3, \dots$ (2) The $i$ th outcome (category) occurs with probability $P_{i}$ , where $i = 1, 2, \dots,$ , $k$ (3)

$\sum_{i = 1}^{k} p_{i} = 1$

Question 11.2

2. Explain in your own words what is meant by a goodness of fit test. (p. 635)

Question 11.3

3. Explain the meaning of the term expected frequency. (Hint: Use the idea of the long-run mean in your answer.) (p. 634)

11.1.3

It is the long-run mean of that random variable after an arbitrarily large number of trials.

Question 11.4

4. State the hypotheses for a $χ^{2}$ goodness of fit test. (p. 634)

PRACTICING THE TECHNIQUES

CHECK IT OUT!

To do	Check out	Topic
Exercises 5–8	Example 1	Identifying a multinomial random variable
Exercises 9–12	Example 2	Finding the expected frequencies
Exercises 13–18	Example 3	Calculating $χ_{data}^{2}$
Exercises 19–22	Example 4	$χ^{2}$ Goodness of fit test: critical-value method
Exercises 23–26	Example 5	$χ^{2}$ Goodness of fit test: p-value method

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For Exercises 5–8, determine whether the random variable is multinomial.

Question 11.5

5. We take a sample of 10 M&M's at random and with replacement and observe the color of each. Let $X = color of the M&M's$ , where the possible values of $X$ are green, blue, yellow, orange, red, and brown.

11.1.5

Multinomial

Question 11.6

6. We select 5 students from a group of 25 statistics students at random and without replacement, and we define our random variable to be $X = student's class$ , where the possible values of $X$ are freshman, sophomore, junior, and senior.

Question 11.7

7. We choose 10 stocks at random and with replacement, and we define our random variable to be $X = the exchange on which the stock is traded$ . The possible values of $X$ are New York Stock Exchange, NASDAQ, London Stock Exchange, and Shenzhen Stock Exchange.

11.1.7

Multinomial

Question 11.8

8. We pick 10 stocks at random and with replacement, and we define our random variable to be $X = the amount that the stock price increased or decreased since the last trading day$ .

For Exercises 9–12, the alternative hypothesis takes the form

$H_{a} : The random variable does not follow the distribution specified in H_{0}$ .

Find the expected frequencies.
Determine whether the conditions for performing the $χ^{2}$ goodness of fit test are met.

Question 11.9

9. $H_{0} : p_{1} = 0.50, p_{2} = 0.25, p_{3} = 0.25; n = 100$

11.1.9

(a) $E_{1} = 50$ , $E_{2} = 25$ , $E_{3} = 25$ (b) Conditions are met.

Question 11.10

10. $H_{0} : p_{1} = 0.2, p_{2} = 0.3, p_{3} = 0.4; p_{4} = 0.1; n = 10$

Question 11.11

11. $H_{0} : p_{1} = 0.9, p_{2} = 0.05, p_{3} = 0.04, p_{4} = 0.01; n = 100$

11.1.11

(a) $E_{1} = 90$ , $E_{2} = 5$ , $E_{3} = 4$ , $E_{4} = 1$ (b) Conditions are not met.

Question 11.12

12. $H_{0} : p_{1} = 0.4, p_{2} = 0.35, p_{3} = 0.10, p_{4} = 0.10, p_{5} = 0.05; n = 200$

For Exercises 13–18, calculate the value of $χ_{data}^{2}$ .

Question 11.13

13.

$O_{i}$	$E_{i}$
10	12
12	12
14	12

11.1.13

0.667

Question 11.14

14.

$O_{i}$	$E_{i}$
15	10
20	25
25	25

Question 11.15

15.

$O_{i}$	$E_{i}$
20	25
30	25
40	30
40	50

11.1.15

7.333

Question 11.16

16.

$O_{i}$	$E_{i}$
8	6
10	8
7	9
5	7

Question 11.17

17.

$O_{i}$	$E_{i}$
1	6
10	6
8	6
0	6
11	6

11.1.17

17.667

Question 11.18

18.

$O_{i}$	$E_{i}$
90	100
100	110
100	90
100	80
110	120

For Exercises 19–22, do the following:

Calculate the expected frequencies and verify that the conditions for performing the $χ^{2}$ goodness of fit test are met.
Find $χ_{crit}^{2}$ for the $χ^{2}$ distribution with the given degrees of freedom. State the rejection rule.
Calculate $χ_{data}^{2}$ .
Compare $χ_{data}^{2}$ with $χ_{crit}^{2}$ . State the conclusion and the interpretation.

Question 11.19

19. $H_{0} : p_{1} = 0.4, p_{2} = 0.3, p_{3} = 0.3; O_{1} = 50, O_{2} = 25, O_{3} = 25;$ level of significance $α = 0.05$

11.1.19

(a) $E_{1} = 40$ , $E_{2} = 30$ , $E_{3} = 30$ ; conditions are met. (b) $χ_{crit}^{2} = χ_{0.05}^{2} = 5.991$ . Reject $H_{0}$ if $χ_{data}^{2} \geq 5.991$ . (c) 4.167 (d) Since $χ_{data}^{2}$ is not $\geq 5.991$ , we do not reject $H_{0}$ . There is insufficient evidence that the random variable does not follow the distribution specified in $H_{0}$ .

Question 11.20

20. $H_{0} : p_{1} = 1 / 3, p_{2} = 1 / 3, p_{3} = 0.3; O_{1} = 40, O_{2} = 30, O_{3} = 20;$ level of significance $α = 0.01$

Question 11.21

21. $H_{0} : p_{1} = 0.4, p_{2} = 0.35, p_{3} = 0.10, p_{4} = 0.10, p_{5} = 0.005; O_{1} = 90, O_{2} = 75, O_{3} = 15, O_{4} = 15, O_{5} = 5;$ level of significance $α = 0.10$

11.1.21

(a) $E_{1} = 80$ , $E_{2} = 70$ , $E_{3} = 20$ , $E_{4} = 20$ , $E_{5} = 10$ ; conditions are met. (b) $χ_{crit}^{2} = χ_{0.10}^{2} = 7.779$ . Reject $H_{0}$ if $χ_{data}^{2} \geq 7.779$ . (c) 6.607 (d) Since $χ_{data}^{2}$ is not $\geq 7.779$ , we do not reject $H_{0}$ . There is insufficient evidence that the random variable does not follow the distribution specified in $H_{0}$ .

Question 11.22

22. $H_{0} : p_{1} = 0.3, p_{2} = 0.2, p_{3} = 0.2, p_{4} = 0.2, p_{5} = 0.1; O_{1} = 63, O_{2} = 42, O_{3} = 40, O_{4} = 38, O_{5} = 17;$ level of significance $α = 0.05$

For Exercises 23–26, do the following:

State the rejection rule for the p-value method, calculate the expected frequencies, and verify that the conditions for performing the $χ^{2}$ goodness of fit test are met.
Calculate $χ_{data}^{2}$
Find the p-value.
Compare the p-value with level of significance $α$ . State the conclusion and the interpretation.

Question 11.23

23. $H_{0} : p_{1} = 0.50, p_{2} = 0.50; O_{1} = 40, O_{2} = 60;$ level of significance $α = 0.05$

11.1.23

(a) Reject $H_{0}$ if the $p$ -value $\leq 0.05$ . $E_{1} = 50$ , $E_{2} = 50$ ; conditions are met. (b) 4 (c) $p$ -value $= 0.0455$ . (d) Since the $p$ -value $\leq 0.05$ , we reject $H_{0}$ . There is evidence that the random variable does not follow the distribution specified in $H_{0}$ .

Question 11.24

24. $H_{0} : p_{1} = 0.50, p_{2} = 0.25; p_{3} = 0.25, O_{1} = 52, O_{2} = 23, O_{3} = 25;$ level of significance $α = 0.10$

Question 11.25

25. $H_{0} : p_{1} = 0.5, p_{2} = 0.25; p_{3} = 0.15, p_{4} = 0.1; O_{1} = 90, O_{2} = 55, O_{3} = 40, O_{4} = 15;$ level of significance $α = 0.10$

11.1.25

(a) Reject $H_{0}$ if the $p$ -value $\leq 0.10$ . $E_{1} = 100$ , $E_{2} = 50$ , $E_{3} = 30$ , $E_{4} = 20$ ; conditions are met. (b) 6.083 (c) $p$ -value $= 0.1076$ . (d) Since the $p$ -value is not $\leq 0.10$ , we do not reject $H_{0}$ . There is insufficient evidence that the random variable does not follow the distribution specified in $H_{0}$ .

Question 11.26

26. $H_{0} : p_{1} = 0.4, p_{2} = 0.2, p_{3} = 0.2, p_{4} = 0.1, p_{5} = 0.1; O_{1} = 90, O_{2} = 45, O_{3} = 40, O_{4} = 15, O_{5} = 10;$ level of significance $α = 0.05$

APPLYING THE CONCEPTS

Question 11.27

27. Sales of Children's Books The market share for sales of children's books in 2013 was as follows (Source: PublishersWeekly.com).

Barnes and Noble	Amazon	Others
23%	20%	57%

Suppose a survey of 1000 children's book sales this year showed 200 sold by Barnes and Noble, 220 sold by Amazon, and 580 sold by all others. Test whether the population proportions have changed, using level of significance $α = 0.05$ .

11.1.27

$H_{0} : p_{Barnes and Noble} = 0.23, p_{Amazon} = 0.20, p_{Others} = 0.57$

$H_{a}$ : The random variable does not follow the distribution specified in $H_{0}$ .

Checking the conditions, the expected frequencies are

$E_{Barnes and Noble} = 230, E_{Amazon} = 200, E_{Others} = 570$

Because none of these expected frequencies is less than one, and none of the expected frequencies is less than five, the conditions for performing the goodness of ft test are met.

$χ_{crit}^{2} = 5.991$ . Reject $H_{0}$ if $χ_{data}^{2} \geq 5.991$ .

$χ_{data}^{2} \approx 6.088$ .

Compare $χ_{data}^{2}$ with $χ_{crit}^{2} \cdot χ_{data}^{2} \approx 6.088$ is $χ_{crit}^{2} = 5.991$ . Therefore, we reject $H_{0}$ . There is evidence that the variable seller does not follow the distribution specified in $H_{0}$ . In other words, there is evidence that the market share for sales of children's books has changed.

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Question 11.28

28. Believing in Angels. Do you believe in angels? A Gallup Poll found that 78% of respondents believed in angels, 12% were not sure or had no opinion, and 10% didn't believe in angels. Suppose that a survey taken this year of 1000 randomly selected people had the following results.

Believe in angels?	Yes	No	Not sure or no opinion
Frequency	820	110	70

Test whether the population proportions have changed, using level of significance $α = 0.05$ .

Question 11.29

29. operating System Market Share. NetMarketShare.com reported that, in 2014, the market share for the desktop operating system market was as follows.

Windows 7	Windows XP	Windows 8	Others
51%	25%	13%	11%

A survey this year of 10,000 randomly selected desktop operating system had the following results.

Windows 7	Windows XP	Windows 8	Others
4500	1500	2000	2000

Test whether the population proportions have changed since 2014, using level of significance $α = 0.05$ .

11.1.29

$H_{0} : p_{Windows 7} = 0.51$ , $p_{Windows xp} = 0.25$ , $p_{Windows 8} = 0.13$ , $p_{Others} = 0.11$ .

$H_{a}$ : The random variable does not follow the distribution specified in $H_{0}$ . Checking the conditions, the expected frequencies are $E_{Windows 7} = 5, 100$ , $E_{Windows xp} = 2, 500$ , $E_{Windows 8} = 1, 300$ , $E_{Others} = 1, 100$ , Because none of these expected frequencies is less than one, and none of the expected frequencies is less than five, the conditions for performing the goodness of ft test are met. $χ_{crit}^{2} = 7.815$ . Reject $H_{0}$ if $χ_{data}^{2} \geq 7.815$ . $χ_{data}^{2} \approx 1583.875$ . Compare $χ_{data}^{2}$ with $χ_{crit}^{2} \cdot χ_{data}^{2} \approx 1583.875$ is $\geq χ_{data}^{2} = 7.815$ . Therefore, we reject $H_{0}$ . There is evidence that the variable operating system does not follow the distribution specified in $H_{0}$ . In other words, there is evidence that the desktop operating systems market share has changed.

Question 11.30

30. Adult Education. The National Center for Education Statistics reported on the percentages of adults who enrolled in personal-interest courses, by the highest education level completed.³ Of these, 8% had less than a high school diploma, 23% had a high school diploma, 32% had some college, 24% had a Bachelor's degree, and 13% had a graduate or professional degree. A survey taken of 200 randomly selected adults who enrolled in personal-interest courses showed the following numbers for the highest education level completed. Test whether the distribution of education levels has changed, using level of significance $α = 0.05$ .

Less than high school	High school diploma	Some college	Bachelor's degree	Graduate or professional degree
12	40	62	54	32

Question 11.31

31. Mall Restaurants. Based on monthly sales data, the International Council of Shopping Centers reported that the proportions of meals eaten at food establishments in shopping malls were as follows: fast food, 30%; food court, 46%; and restaurants, 24%. A survey of 100 randomly selected meals eaten at malls showed that 32 were eaten at fast-food places, 49 were eaten at food courts, and the rest were eaten at restaurants. Test whether the population proportions have changed, using level of significance $α = 0.10$

11.1.31

$H_{0} : p_{fast food} = 0.30$ , $p_{food courts} = 0.46$ , $p_{restaurants} = 0.24$ . $H_{a}$ : The random variable does not follow the distribution specified in $E_{fast food} = 30$ , $E_{food courts} = 46$ , $E_{restaurants} = 24$ . Since none of the expected frequencies is less than 1 and none of the expected frequencies is less than 5, the conditions for performing the $χ^{2}$ goodness of ft test are met. $χ_{crit}^{2} = χ_{0.10}^{2} = 4.605$ . Reject $H_{0}$ if $χ_{data}^{2} \geq 4.605$ . $χ_{data}^{2} = 1.371$ . Since $χ_{data}^{2}$ is not $\geq 4.605$ , we do not reject $H_{0}$ . There is insufficient evidence that the random variable does not follow the distribution specified in $H_{0}$ .

Question 11.32

32. Spinal Cord Injuries. A study found that, of the minority patients who suffered spinal cord injury, 30% had a private health insurance provider, 55.6% used Medicare or Medicaid, and 14.4% had other arrangements.⁴ Suppose that a sample of 1000 randomly selected minority patients with spinal cord injuries found that 350 had a private health insurance provider, 500 used Medicare or Medicaid, and 150 had other arrangements. Test whether the proportions have changed, using level of significance $α = 0.05$ .

Question 11.33

33. University Dining. The university dining service believes there is no difference in student preference among the following four entrees: pizza, cheeseburgers, quiche, and sushi. A sample of 500 students showed that 250 preferred pizza, 215 preferred cheeseburgers, 30 preferred quiche, and 5 preferred sushi. Test, at level of significance $α = 0.01$ , whether or not there is a difference in student preference among the four entrees. (Hint: For the $χ^{2}$ test of no difference among the proportions, the null hypothesis states that all proportions are equal.)

11.1.33

$H_{0} : p_{pizza} = 0.25$ , $p_{cheeseburger} = 0.25$ , $p_{quiche} = 0.25$ , $p_{sushi} = 0.25$ . $H_{a}$ : The random variable does not follow the distribution specified in $H_{0}$ . $E_{p i z z a} = 125$ , $E_{c h e e s e b u r g e r} = 125$ , $E_{q u i c h e} = 125$ , $E_{s u s h i} = 125$ . Since none of the expected frequencies is less than 1 and none of the expected frequencies is less than 5, the conditions for performing the $χ^{2}$ goodness of ft test are met. $χ_{data}^{2} \geq 11.345$ . Reject $H_{0}$ if $χ_{data}^{2} \geq 11.345$ . $χ_{data}^{2} = 377.2$ . Since $χ_{data}^{2} \geq 11.345$ , we reject $H_{0}$ . There is evidence that there is a difference in student preference among the four entries.

Question 11.34

34. ISP Market Share. NetMarketShare.com reported that, in 2014, the worldwide market share for the ISP (Internet service provider) market was as follows.

Comcast	AT&T	Time- Warner	Verizon	Others
7%	7%	5%	5%	76%

A survey this year of 10,000 randomly selected ISPs provided the following results.

Comcast	AT&T	Time- Warner	Verizon	Others
900	800	600	700	7000

Test whether the population proportions have changed since 2014, using level of significance $α = 0.01$ .

Question 11.35

35. Refer to the previous exercise. What if the observed frequency for Comcast had been larger and the observed frequency for Others had been smaller, by some unknown amount? Describe how this would affect the following:

The hypotheses
The expected frequencies
$χ_{data}^{2}$
The p-value
The conclusion

11.1.35

(a) Stay the same (b) Stay the same (c) Increases (d) Decreases (e) Stays the same