Question

One True Love by Educational Level In Data 2.1 on page 48 , we introduce a study in which people were asked whether they agreed or disagreed with the statement that there is only one true love for each person. Table 7.29 gives a two-way table showing the answers to this question as well as the education level of the respondents. A person's education is categorized as HS (high school degree or less), Some (some college), or College (college graduate or higher). Is the level of a person's education related to how the person feels about one true love? If there is a significant association between these two variables, describe how they are related. $$ \begin{array}{l|rrr|r} \hline & \text { HS } & \text { Some } & \text { College } & \text { Total } \\\ \hline \text { Agree } & 363 & 176 & 196 & 735 \\ \text { Disagree } & 557 & 466 & 789 & 1812 \\ \text { Don't know } & 20 & 26 & 32 & 78 \\ \hline \text { Total } & 940 & 668 & 1017 & 2625 \\ \hline \end{array} $$

Short answer

The solution requires to conduct a chi-square independence test. After calculating the expected frequencies, the chi-square test statistic, and the P-value, if the P-value is less than 0.05, then there is a significant association between level of education and belief in one 'true love'. The nature of the relationship will be further interpreted by examining the differences between observed and expected frequencies.

Step-by-step solution

Calculate Expected Frequencies

Under the assumption of independence, the expected frequency for each cell in the table is calculated by the formula: \( E_{i,j} = (n_{i} . n_{j}) / N \) where \( n_i \) and \( n_j \) are the row and column totals respectively, and N is the grand total. Calculate the expected frequencies for each cell using this formula.

Compute the Chi-Square Test Statistic

After calculating the expected frequencies, compute the chi-square test statistic using the formula: \( χ^2 = Σ [ (O_{i,j} - E_{i,j})^2 / E_{i,j} ] \) where \( O_{i,j} \) and \( E_{i,j} \) are the observed and expected frequencies respectively. Take the sum of these values for all cells.

Calculate the P-Value

Use the calculated chi-square test statistic and the degrees of freedom to find the P-value. The degrees of freedom for this test is given by (r-1)(c-1), where r is the number of rows and c is the number of columns in the contingency table. If the P-value is less than 0.05, then reject the null hypothesis (that the level of education and belief in one 'true love' are independent) and conclude that there is a significant association between the two.

Interpret Results

If the P-value is less than 0.05, this indicates that the level of education and belief in one 'true love' are not independent – there is a significant association. To further understand the relationship, examine the differences between observed and expected frequencies. Higher observed frequencies than expected in certain cells indicate a stronger relationship for those categories.