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Chapter 11: Q7CQQ (page 533)

Questions 6鈥�10 refer to the sample data in the following table, which describes the fate of the passengers and crew aboard the Titanic when it sank on April 15, 1912. Assume that the data are a sample from a large population and we want to use a 0.05 significance level to test the claim that surviving is independent of whether the person is a man, woman, boy, or girl.


Men

Women

Boys

Girls

Survived

332

318

29

27

Died

1360

104

35

18

What distribution is used to test the stated claim (normal, t, F, chi-square, uniform)?

Short Answer

Expert verified

The chi-square distribution is used to test the given claim.

Step by step solution

01

Given information

A contingency table is constructed that shows the number of passengers who survived/died according to whether they were male, female, boy or girl.

02

Distribution of the test statistic

It is required to test the claim that the survival of the passenger is independent of whether the person is a man, woman, boy or girl.

Thus, it is a test of the independence of attributes that involves nominal variables or variables that consist of categories.

Here, the row variable consists of labels 鈥渟urvived鈥� and 鈥渄ied鈥� and the column variable consists of labels 鈥渕en鈥�, 鈥渨omen鈥�, 鈥渂oys鈥� and 鈥済irls鈥�.

Such types of tests that involve categorical variables are carried out using the chi-square distribution.

Moreover, the other hypotheses tests involving normal distribution, t-distribution and F-distribution are applied only when the variables are quantitative and not qualitative.

Uniform distribution is not used to conduct hypothesis tests.

Therefore, the distribution of the test statistic to test the given claim is the chi-square distribution.

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Most popular questions from this chapter

Forward Grip Reach and Ergonomics When designing cars and aircraft, we must consider the forward grip reach of women. Women have normally distributed forward grip reaches with a mean of 686 mm and a standard deviation of 34 mm (based on anthropometric survey data from Gordon, Churchill, et al.).

a. If a car dashboard is positioned so that it can be reached by 95% of women, what is the shortest forward grip reach that can access the dashboard?

b. If a car dashboard is positioned so that it can be reached by women with a grip reach greater than 650 mm, what percentage of women cannot reach the dashboard? Is that percentage too high?

c. Find the probability that 16 randomly selected women have forward grip reaches with a mean greater than 680 mm. Does this result have any effect on the design?

Equivalent Tests A\({\chi ^2}\)test involving a 2\( \times \)2 table is equivalent to the test for the differencebetween two proportions, as described in Section 9-1. Using the claim and table inExercise 9 鈥淔our Quarters the Same as $1?鈥� verify that the\({\chi ^2}\)test statistic and the zteststatistic (found from the test of equality of two proportions) are related as follows:\({z^2}\)=\({\chi ^2}\).

Also show that the critical values have that same relationship.

One Big Bill or Many Smaller Bills In a study of the 鈥渄enomination effect,鈥� 150 women in China were given either a single 100 yuan bill or a total of 100 yuan in smaller bills. The value of 100 yuan is about $15. The women were given the choice of spending the money on specific items or keeping the money. The results are summarized in the table below (based on 鈥淭he Denomination Effect,鈥� by Priya Raghubir and Joydeep Srivastava, Journal of Consumer Research, Vol. 36). Use a 0.05 significance level to test the claim that the form of the 100 yuan is independent of whether the money was spent. What does the result suggest about a denomination effect?

Spent the Money

Kept the Money

Women Given a Single 100-Yuan Bill

60

15

Women Given 100 Yuan in Smaller Bills

68

7

Exercises 1鈥�5 refer to the sample data in the following table, which summarizes the last digits of the heights (cm) of 300 randomly selected subjects (from Data Set 1 鈥淏ody Data鈥� in Appendix B). Assume that we want to use a 0.05 significance level to test the claim that the data are from a population having the property that the last digits are all equally likely.

Last Digit

0

1

2

3

4

5

6

7

8

9

Frequency

30

35

24

25

35

36

37

27

27

24

If using a 0.05 significance level to test the stated claim, find the number of degrees of freedom.

A case-control (or retrospective) study was conductedto investigate a relationship between the colors of helmets worn by motorcycle drivers andwhether they are injured or killed in a crash. Results are given in the table below (based on datafrom 鈥淢otorcycle Rider Conspicuity and Crash Related Injury: Case-Control Study,鈥� by Wellset al., BMJ USA,Vol. 4). Test the claim that injuries are independent of helmet color. Shouldmotorcycle drivers choose helmets with a particular color? If so, which color appears best?

Color of helmet


Black

White

Yellow/Orange

Red

Blue

Controls (not injured)

491

377

31

170

55

Cases (injured or killed)

213

112

8

70

26
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