Choose either 2 ¡Á 2 contingency table shown below (males or females). Research question: At
¦Á = .005, is smoking independent of race? (Smoking rates are from Statistical Abstract of the
United States, 2001, pp. 16 and 12, applied to hypothetical samples of 500.) Smoking
Smoking by Race for Males Aged 18¨C24
Race Smoker Nonsmoker Row Total
White 145 280 425
Black 15 60 75
Col Total 160 340 500
Smoking by Race for Females Aged 18¨C24
Race Smoker Nonsmoker Row Total
White 116 299 415
Black 7 78 85
Col Total 123 377 500
To determine if smoking is independent of race at a significance level of α = 0.005, we need to perform a chi-square test of independence.
To calculate the expected frequencies for each cell in the contingency table, we use the formula:
Expected Frequency = (Row Total × Column Total) / Total Sample Size
First, let's calculate the expected frequencies for the Male 2 × 2 contingency table:
Expected frequency for Male White Smokers = (425 × 160) / 500 = 136
Expected frequency for Male White Nonsmokers = (425 × 340) / 500 = 289
Expected frequency for Male Black Smokers = (75 × 160) / 500 = 24
Expected frequency for Male Black Nonsmokers = (75 × 340) / 500 = 51
Now, let's calculate the expected frequencies for the Female 2 × 2 contingency table:
Expected frequency for Female White Smokers = (415 × 123) / 500 = 102
Expected frequency for Female White Nonsmokers = (415 × 377) / 500 = 313
Expected frequency for Female Black Smokers = (85 × 123) / 500 = 21
Expected frequency for Female Black Nonsmokers = (85 × 377) / 500 = 64
Next, we can set up the observed and expected frequency tables:
Male:
Smoker Nonsmoker Total
White 145 280 425
Black 15 60 75
Total 160 340 500
Smoker Nonsmoker Total
White 136 289 425
Black 24 51 75
Total 160 340 500
Female:
Smoker Nonsmoker Total
White 116 299 415
Black 7 78 85
Total 123 377 500
Smoker Nonsmoker Total
White 102 313 415
Black 21 64 85
Total 123 377 500
Now, we can perform the chi-square test of independence using the chi-square formula:
χ^2 = Σ{( (Observed frequency - Expected frequency)^2 ) / Expected frequency}
For both the Male and Female contingency tables, we calculate the chi-square value using this formula.
Once we have the chi-square value, we can compare it to the critical value from the chi-square distribution table (with degrees of freedom equal to (number of rows - 1) × (number of columns - 1)). If the chi-square value exceeds the critical value, we reject the null hypothesis of independence.
Therefore, to answer the research question, we need to calculate the chi-square values for both the Male and Female contingency tables and compare them to the critical value at α = 0.005 level of significance. If the chi-square values exceed the critical value, we can conclude that smoking rates are not independent of race.