No smoking: The General Social Survey conducted a poll of 691 adults in which the subjects were asked whether they agree that the government should prohibit smoking in public places. In addition, each was asked how many people lived in his or her household. The results are summarized in the following contingency table. | Household Size | 1 | 2 | 3 | 4 | 5 | | :------------- | :-- | :-- | :-- | :-- | :-- | | Agree | 66 | 107 | 37 | 29 | 31 | | No Opinion | 27 | 55 | 30 | 19 | 29 | | Disagree | 52 | 80 | 47 | 50 | 32 | Test the hypothesis of independence. Use the α = 0.05 level of significance and the P-value method with the TI-84 Plus calculator. What do you conclude? (a) State the null and alternate hypotheses. H0: Household size and opinion are independent. H1: Household size and opinion are dependent.
Answer (1)
To solve this problem, we need to test the hypothesis of independence between 'Household Size' and 'Opinion' regarding the prohibition of smoking in public places. We'll use a chi-square test for independence with the contingency table data provided.
Step-by-Step Solution
State the Hypotheses:
Null Hypothesis H 0 : Household size and opinion are independent.
Alternative Hypothesis H 1 : Household size and opinion are dependent.
Set the Significance Level:
The significance level α is given as 0.05.
Enter the Data into the TI-84 Plus Calculator:
Go to [STAT] -> [EDIT] and enter the contingency table for 'Agree', 'No Opinion', and 'Disagree' across different household sizes into the lists.
Perform the Chi-Square Test:
Access the Chi-Square Test feature by pressing [STAT], then selecting [TESTS], and choosing χ 2 -Test.
Input the observed values and have the calculator compute the expected frequencies, chi-square statistic, and p-value.
Calculate the Test Statistic and P-value:
The calculator will output the chi-square statistic χ 2 and the p-value.
Decision Based on P-value:
If the p-value is less than or equal to the significance level (0.05), reject the null hypothesis H 0 .
If the p-value is greater than 0.05, do not reject H 0 .
Conclusion:
Based on the obtained p-value, make a conclusion about the independence of household size and opinion. If H 0 is rejected, it suggests that household size and opinion are dependent; otherwise, they are independent.
Note
The detailed calculations, including expected frequencies and the chi-square statistic, will depend on the data entered and the output provided by the calculator. You should follow these steps on the calculator to get the exact result for your specific data.
