Question

Gender and Frequency of "Liking" Content on Facebook Exercise 7.48 introduces a study about users of social networking sites such as Facebook. Table 7.37 shows the frequency of users "liking" content on Facebook, with the data shown by gender. Does the frequency of "liking" depend on the gender of the user? Show all details of the test. $$ \begin{array}{l|rr|r} \hline \downarrow \text { Liking/Gender } \rightarrow & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { Every day } & 77 & 142 & 219 \\ \text { 3-5 days/week } & 39 & 54 & 93 \\ \text { 1-2 days/week } & 62 & 69 & 131 \\ \text { Every few weeks } & 42 & 44 & 86 \\ \text { Less often } & 166 & 182 & 348 \\ \hline \text { Total } & 386 & 491 & 877 \end{array} $$

Short answer

The complete answer requires calculation and depends on the results of those calculations. After calculating expected frequencies, chi-square test statistic, degrees of freedom and comparing the p-value to the significance level, one can determine whether the liking behavior on Facebook depends on gender.

Step-by-step solution

Calculations of Expected Frequencies

Under the null hypothesis of gender and liking behavior being independent, the expected frequency of a particular combination is calculated as (Row Total * Column Total) / Grand Total. Do this for every cell in the table to get the expected frequencies.

Chi-Square Test Statistic

The test statistic for the chi-square test is given by the sum over all cells: \((Observed-Expected)^2/Expected\). Compute this value.

Degrees of Freedom and P-value

The degrees of freedom for this test is given by (Number of Rows - 1) * (Number of Columns - 1). Use this degree of freedom and the test statistic computed in Step 2 to find the p-value, referencing the chi-square distribution table.

Conclusion

If the p-value is less than the significance level (generally .05), reject the null hypothesis and conclude that the frequency of 'liking' does indeed depend on the gender. If not, there is not enough evidence to say that liking behaviors on Facebook depend on gender.

Key concepts

Expected Frequencies

When performing a chi-square test, a crucial step is calculating the expected frequencies. These figures represent the expected counts in each category if the null hypothesis were true, meaning there is no association between the variables.

In the context of our exercise, the expected frequency for each cell is found by multiplying the total number of observations in that row by the total number of observations in that column, and then dividing by the grand total of all observations. For example, to calculate the expected frequency of males 'liking' content every day, one would use the formula: \( \frac{\text{Total Every day} \times \text{Total Male}}{\text{Grand Total}} \).

It's essential to compute these expected values accurately as they are used as a benchmark to measure how far off the observed frequencies are from what we would expect to see if the variables were truly independent.

Test Statistic

The test statistic in a chi-square test quantifies how much the observed frequencies deviate from the expected frequencies. It’s a measure of the disparity between what was actually observed in the data and what we would anticipate under the null hypothesis of no association.

To calculate the chi-square test statistic, each cell's observed value is compared to its expected value using the formula: \( (\text{Observed} - \text{Expected})^2 / \text{Expected} \).

After summing these values for all cells, you get a single number that indicates the overall difference. The larger the test statistic, the more evidence there is of discrepancy between the observed and expected data, suggesting a possible association between the variables in question.

Degrees of Freedom

Degrees of freedom in statistics can be thought of as the number of values that can vary without violating given constraints. They play a key role in determining the critical value against which the test statistic is compared in a chi-square test.

The formula for calculating degrees of freedom in a contingency table like our exercise is: \( (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1) \). In this scenario, with a 2x5 contingency table (2 genders and 5 frequency categories), the chi-square test has \( (2-1) \times (5-1) = 4 \) degrees of freedom.

This informs us about the chi-square distribution that we will reference to interpret the test statistic and ultimately reach a conclusion about our hypothesis.

P-value

The p-value is a probability that provides a measure of the evidence against the null hypothesis provided by the data. Specifically, it's the probability of observing a test statistic at least as extreme as the one computed, assuming the null hypothesis is true.

In our sample exercise, once the chi-square statistic is calculated, the corresponding p-value can be found by looking up this statistic in the chi-square distribution table, with the degrees of freedom we calculated.

A smaller p-value means stronger evidence against the null hypothesis. Typically, if the p-value is below the predefined level of significance, which is often 0.05, we would reject the null hypothesis, indicating that there is a statistically significant association between the variables.

Statistical Significance

Statistical significance is used to determine whether the observed results are due to chance or if they reflect a true effect. In hypothesis testing, statistical significance is assessed by comparing the p-value to a predetermined significance level, often set at 0.05.

If the p-value is less than the chosen significance level, we reject the null hypothesis, suggesting that the results are unlikely to have occurred by random chance, thus inferring a statistically significant relationship between the variables.

In the context of our exercise, if the p-value is below 0.05, we would conclude that the frequency of 'liking' content on Facebook is dependent on gender. If the p-value is above 0.05, we lack sufficient evidence to claim a relationship, and the null hypothesis that 'liking' behavior is independent of gender would not be rejected.