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Although consumers often want to touch products before purchasing them, they generally prefer that others have not touched products they would like to buy. Can another person touching a product create a positive reaction? Subjects were given instructions to contact a sales associate at a university bookstore who would provide them with a shirt to try on. When meeting the sales associate, subjects were told there was only one shirt left and it was being tried on by another "customer." The other customer trying on the shirt was a confederate of the experimenter and was either an attractive, well-dressed professional female model or an average-looking female college student wearing jeans and a T-shirt. Subjects, who were either males or females, saw the confederate leaving the dressing room where the shirt was left for them to try on. There was also a control group of subjects who were handed the shirt directly off the rack by the sales associate. Thus, there were five treatments: male subjects seeing a model, female subjects seeing a model, male subjects seeing a college student, female subjects seeing a college student, and the control group. Subjects evaluated the product on five dimensions, each dimension on a seven-point scale, with the five scores then averaged to give the subject's evaluation measure, with higher numbers indicating a more positive evaluation. Here are the sample sizes, means, and standard deviations for the five groups: \({ }^{15}\) $$ \begin{array}{lccc} \hline \text { Treatment Group } & n & \mathrm{x}^{-} \bar{x} & \boldsymbol{s} \\\ \hline \text { Males seeing a model } & 22 & 5.34 & 0.87 \\ \hline \text { Males seeing a student } & 23 & 3.32 & 1.21 \\ \hline \text { Females seeing a model } & 24 & 4.10 & 1.32 \\ \hline \text { Females seeing a student } & 23 & 3.50 & 1.43 \\ \hline \text { Controls } & 27 & 4.17 & 1.50 \\ \hline \end{array} $$ (a) Verify that the sample standard deviations allow the use of ANOVA to compare the population means. What do the means suggest about the effect of the subject's sex and the attractiveness of the confederate on the evaluation of the product? (b) The paper reports the ANOVA \(F=8.30\). What are the degrees of freedom for the ANOVA \(F\) statistic and the \(P\)-value? State your conclusions.

Step by step solution

01

Assess Homogeneity of Variances

To verify if ANOVA can be used, we need the standard deviations to be relatively similar across groups. The given standard deviations are 0.87, 1.21, 1.32, 1.43, and 1.50 for the different groups. The largest standard deviation (1.50) is less than twice the smallest (0.87), so homogeneity of variances assumption is satisfied.

02

Interpret the Means

The means for each group are 5.34, 3.32, 4.10, 3.50, and 4.17 respectively. This suggests that the most positive evaluation (5.34) comes from males seeing a model. Comparatively, males seeing a student had the lowest mean (3.32). Other groups had intermediate evaluations. Attraction may positively influence product evaluation, especially for males.

03

ANOVA Degrees of Freedom

The degrees of freedom for the between-groups variation is the number of groups minus one: \(df_1 = 5 - 1 = 4\). The degrees of freedom for the within-group variation is the sum of each group's sample size minus the number of groups: \(df_2 = 22 + 23 + 24 + 23 + 27 - 5 = 114\).

04

Determine P-value and Conclusion

Given ANOVA \(F=8.30\), degrees of freedom \((4, 114)\), and knowing that a typical significance level is 0.05, we look up \(F\)-table or use a calculator. An \(F\)-value of 8.30 with these degrees of freedom typically implies \(P < 0.05\), suggesting rejecting the null hypothesis of no difference between means. Thus, attractiveness and subject's sex significantly influence product evaluations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variance Homogeneity

Before conducting an ANOVA test, it's critical to check for the homogeneity of variances among the groups being compared. This condition, also known as homoscedasticity, implies that the variances should be fairly similar across all groups. We assess this by looking at the sample standard deviations provided in the data. In this scenario, the standard deviations for the groups are given as 0.87, 1.21, 1.32, 1.43, and 1.50. A common rule of thumb for assessing variance homogeneity is that the largest standard deviation should not be more than twice the smallest. Here, the largest standard deviation is 1.50, and the smallest is 0.87. This ratio is less than two, indicating that the assumption of variance homogeneity is satisfied.

When variances are homogeneous, it means that variations within each group are consistent, which is a necessary condition to proceed with ANOVA. This ensures that differences observed between treatment group means are due to the effect of treatments rather than inconsistencies in variation within groups. If variances were not homogeneous, it could potentially lead to incorrect conclusions being drawn from the ANOVA test.

Product Evaluation

In the context of the experiment, product evaluation plays a crucial role. Participants in the study were asked to evaluate a shirt they tried on after seeing a confederate wearing it. The evaluation was based on five dimensions and recorded on a seven-point scale. Each subject's scores on these dimensions were then averaged to provide an overall evaluation score.

This evaluation process is fundamental as it provides quantifiable data that reflects participants' perceptions about the product after observing the confederate. These evaluations are indicative of how the attractiveness and gender of the confederate might influence perceptions and attitudes towards the product. For instance, the mean scores from the data suggest that males gave higher evaluations when the shirt was last worn by an attractive model compared to an average-looking student. Meanwhile, the females' scores showed less variation, suggesting different influencing factors for product evaluation based on observer gender.

Experimental Design

In this study, the experimental design used is essential for understanding how different variables affect the outcome. A controlled experimental environment was created where subjects were exposed to varying conditions – seeing either an attractive, well-dressed model or an average-looking student before evaluating a product. There was also a control group where subjects received the product directly from a sales associate.

This kind of design helps isolate the effects of 'who wore the product last' on the evaluation scores. It incorporates both between-subject factors (the person trying the product before the subject) and within-subject controls (the method of shirt presentation).

The carefully balanced design between different gender and observer interactions allows researchers to draw conclusions about how the attractiveness of the individual previously wearing the product affects the subject's evaluation. This setup reduces potential bias and provides a robust data set for statistical analysis through ANOVA.

Sample Means

Sample means are vital in analyzing the results of the ANOVA test. They provide an average score for each group and are crucial for comparing the overall product evaluations between different treatment groups.

In the experiment provided, the sample means are 5.34 for males seeing a model, 3.32 for males seeing a student, 4.10 for females seeing a model, 3.50 for females seeing a student, and 4.17 for the control group. These differences in means highlight how varying conditions impact evaluations.

The comparison of these means indicates patterns in how the attractiveness of the confederate and the observer's gender influence product evaluation. For instance, the higher mean for males who saw a model suggests a positive bias toward products associated with attractive wearers. Understanding these means involves considering both their direct values and their implications for the superiority of certain conditions over others, as revealed by the ANOVA results.