Question

Does Red Increase Men's Attraction to Women? Exercise 1.99 on page 44 described a study \(^{46}\) which examines the impact of the color red on how attractive men perceive women to be. In the study, men were randomly divided into two groups and were asked to rate the attractiveness of women on a scale of 1 (not at all attractive) to 9 (extremely attractive). Men in one group were shown pictures of women on a white background while the men in the other group were shown the same pictures of women on a red background. The results are shown in Table 6.14 and the data for both groups are reasonably symmetric with no outliers. To determine the possible effect size of the red background over the white, find and interpret a \(90 \%\) confidence interval for the difference in mean attractiveness rating.

Short answer

Detailed interpretation of the answer will depend on the calculated confidence interval, which we don't have due to the absence of numerical data. If the interval contains zero, it suggests the color of the background is not statistically significant in influencing men's ratings. If not, it means there's evidence to suggest that the color red does (positive difference) or does not (negative difference) increase men's attraction, with 90% confidence.

Step-by-step solution

Extract Key Data

First, extract key information from the provided data. This includes getting the sample sizes for both groups (n1 and n2), the mean attractiveness rating for each group (\(\bar{x}_1\) and \(\bar{x}_2\)), and the standard deviation for each group (s1 and s2).

Calculate the Standard Error

Calculated the standard error (SE) using the formula \(\mathrm{SE} = \sqrt{\frac{s1^2}{n1} + \frac{s2^2}{n2}}\). This represents the standard deviation of the sampling distribution of the differences between the means.

Calculate the t Score

Calculate the t-score using the formula \(t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE}\). This informs us about the number of standard errors that the difference between our sample means is from zero.

Determine the Critical Value

Based on p% (here 90%) and the degrees of freedom (df = n1 + n2 - 2), locate the critical t value from the t-distribution table.

Formulate Confidence Interval

The confidence interval is calculated as \((\bar{x}_1 - \bar{x}_2) \pm (t_{\text{critical}} \times SE)\). The calculated interval gives the range in which we can be 90% sure that the true mean difference in attractiveness ratings lies.

Interpret the Confidence Interval

If the confidence interval contains zero, the difference in attractiveness ratings between the two backgrounds is not statistically significant at the 90% confidence level. If zero is not in the interval, we can claim with 90% confidence that the color red does (or does not, depending on the sign of the difference) increase men's attraction to women.