# Chi-square test of independence, goodness of fit

1) Suppose we have a multinomial population with four categories: A, B, C, and D. The null hypothesis is that the proportion of items is the same in every category. The null hypothesis is

H : p = p = p = p = .25

º ª b c d

A sample of size 300 yielded the following results.

A:85 B:95 C:50 D:70

Use a = .05 to determine whether H should be rejected. What is the p-value?

º

2) The National Sleep Foundation used a survey to determine whether hours of sleeping per night are independent of age. The following show the hours of sleep on weeknights for a sample of individuals age 49 and younger and for a sample of individuals age 50 and older.

Hours of sleep

Age Fewer than 6 6 to 6.9 7 to 7.9 8 or more Total

49 or younger 38 60 77 65 240

50 or older 36 57 75 92 260

a. Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. Use a = .05. What is the p-value, and what is your conclusion?

b. What is your estimate of the percentage of people who sleep fewer than 6 hours, 6 to 6.9 hours, 7 to 7.9 hours, and 8 or more hours on weeknights?

3) The U.S. Bureau of Labor Statistics collected data on the occupation of workers 25 to 64 years old. The following table shows the number of male and female workers (in thousands) in each occupation category (Statistical Abstract of the United States: 2002).

Occupation Male Female

Managerial/professional 19079 19021

Tech./Sales/Administrative 11079 19315

Service 4977 7947

Precision production 11682 1138

Operators/fabricators/labor 10576 3482

Farming/forestry/fishing 1838 514

See attached file for full problem description.

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Answers to 3 questions based on Chi-square test of independence, goodness of fit.