91Ó°ÊÓ

The authors of the paper "The Empowering (Super) Heroine? The Effects of Sexualized Female Characters in Superhero Films on Women" (Sex Roles [2015]: \(211-220\) ) were interested in the effect on female viewers of watching movies in which female heroines were portrayed in roles that focus on their sex appeal. They carried out an experiment in which female college students were assigned at random to one of two experimental groups. The 23 women in one group watched 13 minutes of scenes from the X-Men film series and then responded to a questionnaire designed to measure body esteem. Lower scores on this measure correspond to lower body satisfaction. The 29 women in the other group (the control group) did not watch any video prior to responding to the questionnaire measuring body esteem. For the women who watched the \(X\) -Men video, the mean body esteem score was 4.43 and the standard deviation was \(1.02 .\) For the women in the control group, the mean body esteem score was 5.08 and the standard deviation was \(0.98 .\) For purposes of this exercise, you may assume that the distribution of body esteem scores for each of the two treatments (video and control) is approximately normal. a. Construct and interpret a \(90 \%\) confidence interval for the difference in mean body esteem score for the video treatment and the no video treatment. b. Do you think that watching the video has an effect on mean body esteem score? Explain.

Step by step solution

01

Find sample sizes, means, and standard deviations for both groups

Sample size and statistics for both groups are given in the problem: - Video treatment group (Group 1): n1 = 23, mean1 = 4.43, sd1 = 1.02 - Control group (Group 2): n2 = 29, mean2 = 5.08, sd2 = 0.98

02

Calculate the standard error of the difference in means

We will calculate the standard error (SE) of the difference in means using the formula: SE_diff = \(\sqrt{\frac{sd1^2}{n1} + \frac{sd2^2}{n2}}\) SE_diff = \(\sqrt{\frac{1.02^2}{23} + \frac{0.98^2}{29}} = 0.305\)

03

Find the critical value for a 90% confidence interval

As this is a two-tailed test, we will use a 90% confidence interval, which leaves 5% in each tail. Using a t-distribution table or calculator, we look for the critical value corresponding to 0.05 in the upper tail, with degrees of freedom equal to the smallest of n1-1 and n2-1 (since we are working with small samples), so df = min(23-1,29-1) = 22. The critical t-value (t_critical) is 1.717.

04

Calculate the margin of error (ME)

Margin of error (ME) = t_critical × SE_diff ME = 1.717 × 0.305 = 0.523

05

Construct the confidence interval

The 90% confidence interval for the difference in mean body esteem scores between the video treatment and the control group is: (Mean_diff - ME, Mean_diff + ME) Mean_diff = mean1 - mean2 = 4.43 - 5.08 = -0.65 The 90% CI: (-0.65 - 0.523, -0.65 + 0.523) = (-1.173, -0.127) a. The 90% confidence interval for the difference in mean body esteem score between the video treatment group and the control group is (-1.173, -0.127).

06

Analyze the effect of the video on mean body esteem score

b. Since the 90% confidence interval for the difference in mean body esteem scores between the video and control groups does not contain 0, we have evidence at α = 0.10 level to suggest that there is a statistically significant difference between the two groups. In this case, watching the video appears to have a negative effect on mean body esteem score, as the confidence interval indicates a decrease in the mean body esteem score for those who watched the video compared to those who did not.

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Key Concepts

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

Sampling Distribution

In the context of statistics, the term sampling distribution refers to the probability distribution of a given statistic based on a random sample. In simpler terms, it is the range of values that a statistic, such as a sample mean, can take if different samples are taken from the same population. In our exercise, the mean body esteem scores from two different groups represent two sampling distributions, one for each group.

Understanding sampling distributions is crucial because they form the basis for various types of statistical inferences, including the construction of confidence intervals. By recognizing that these distributions can vary from sample to sample, we can quantify the uncertainty inherent in our estimates.

Standard Error Calculation

The standard error (SE) measures the amount of variability or dispersion in a sampling distribution. Specifically, it represents the standard deviation of the sampling distribution of a statistic, which in our case is the difference in mean body esteem scores between the two groups.

To calculate the standard error of the mean, the formula used incorporates the standard deviations of the individual groups and their respective sample sizes, as seen in the solution step 2. Lower standard error implies that the sample mean is a more precise estimate of the population mean, which is key in assessing the reliability of statistical conclusions.

T-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails. It arises when estimating the mean of a normally distributed population when the sample size is small and the population standard deviation is unknown.

In our exercise, since we have small sample sizes and we estimate the population parameters, we use the t-distribution to determine the critical value for constructing the confidence interval. As the sample size increases, the t-distribution approaches the normal distribution.

Margin of Error

The margin of error (ME) quantifies the range of error around a sample statistic, such as a sample mean. It tells us how far we can expect the true population parameter to be from our sample estimate. The margin of error includes the critical value from the appropriate distribution and the standard error of the statistic.

In the solution steps 4 and 5, the margin of error was calculated using the critical value from the t-distribution and the standard error of the difference in means. This margin of error was then used to construct the confidence interval, which provides a range in which the true difference in population means is likely to fall.

Statistical Significance

The concept of statistical significance is used to determine whether the observed effect in our sample is unlikely to have occurred due to chance, assuming the null hypothesis is true. By convention, a result is considered statistically significant if the probability of the result occurring by chance is below a pre-determined threshold (the alpha level, \( \alpha \) ).

In the final step of the solution, the confidence interval did not include zero, indicating that there is a statistically significant difference in mean body esteem scores between the two groups at the \( \alpha = 0.10 \) level. This signifies that the observed difference in means is unlikely to be due to random variability alone, suggesting a genuine effect of the video on body esteem.