91Ó°ÊÓ

The majority of clothing retailers use mannequins to display their merchandise, with approximately one-third displaying mannequins with a head and two-thirds displaying mannequins without a head. Researchers recruited 126 female participants and assigned them to one of the two mannequin styles (head/headless). Participants were asked to imagine that they wanted to buy a new dress and told that they had gone to a named store to make their purchase. They then viewed the dress displayed on a mannequin (head or headless) and asked whether or not they would buy the dress. Of the 53 participants viewing the dress displayed on a mannequin with a head, 18 indicated they would buy the dress, while only 10 of the 53 participants viewing the headless mannequin indicated they would buy the dress. \({ }^{25}\) (a) Is there good evidence that the proportion of women who would buy the dress differs between those who viewed the dress displayed on a mannequin with or without a head? (b) Based on this study, do you think it is a good idea for most manufacturers to use display mannequins without a head?

Step by step solution

01

Calculate Sample Proportions

Calculate the proportion of participants in each group who decided to buy the dress. For mannequins with a head, the proportion is \( p_1 = \frac{18}{53} \). For mannequins without a head, the proportion is \( p_2 = \frac{10}{53} \).

02

Calculate Pooled Proportion

Combine the data of both groups to calculate the overall proportion of individuals who decided to buy the dress. This is the pooled proportion \( p = \frac{18 + 10}{53 + 53} = \frac{28}{106} \).

03

Calculate Standard Error

The standard error of the difference in sample proportions is calculated using the formula \( SE = \sqrt{p(1-p)\left(\frac{1}{53} + \frac{1}{53}\right)} \), where \( p \) is the pooled proportion.

04

Calculate Z-Score

The Z-score can be calculated to determine the significance of the observed difference. \( Z = \frac{p_1 - p_2}{SE} \). This will tell us how many standard deviations apart our sample proportions are, compared to what we would expect if there were no difference.

05

Determine Significance and Conclusion

Using the Z-score, determine the p-value by comparing it with standard normal distribution values. If the p-value is less than a standard significance level (such as 0.05), it suggests a significant difference between the two groups.

06

Answer Important Questions

(a) If the p-value is less than 0.05, we have evidence that the proportion of women who would buy the dress differs based on the type of mannequin they viewed. (b) If the proportion buying from mannequins with heads is significantly higher, it may indicate that it's preferable for manufacturers to use mannequins with heads; otherwise, the opposite might be true.

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Key Concepts

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

Sample Proportions

In statistical analysis, the concept of sample proportions plays a crucial role. When conducting experiments or studies, we often work with samples because it's impractical to study an entire population. In this case, sample proportions help us understand the behavior of a small group to infer about the larger group.
For example, in the exercise with mannequins, researchers determined how many participants from each group (mannequin with a head versus headless) were willing to buy a dress.

• For mannequins with a head, the sample proportion is calculated as \( p_1 = \frac{18}{53} \), which is approximately 0.34.
• For mannequins without a head, the sample proportion is \( p_2 = \frac{10}{53} \), around 0.19.

These proportions provide a snapshot of each group's decision outcomes. By calculating and comparing these proportions, researchers can identify if there is a notable difference in preferences between the two mannequin types.

Pooled Proportion

The pooled proportion is another essential concept that combines data from different groups to analyze overall trends. This value gives a comprehensive overview of the study's findings by aggregating the data.
To find the pooled proportion in our example, researchers combined the results of both mannequin groups to calculate the overall proportion of participants who would buy the dress:

• The pooled proportion \( p \) is \( \frac{28}{106} \), approximately 0.26.

This calculation accounts for the whole sample, offering a general perspective and enabling a clear comparison between the two groups.
The pooled proportion is particularly useful when testing hypotheses about differences between sample proportions. It helps in deriving a single proportion that can be used in further statistical calculations, like the standard error.

Standard Error

The standard error (SE) measures the variability or the standard deviation of the sampling distribution of a statistic, most often the mean or proportion. It indicates how much a sample proportion might fluctuate from the true population proportion due to random sampling.
In our context, we calculate the SE of the difference between two sample proportions using the formula:

• \( SE = \sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \)

Where \( p \) is the pooled proportion, \( n_1 \) and \( n_2 \) are the sample sizes of the two groups.
The standard error helps determine the reliability of the estimate of the population parameters, facilitating decision-making about whether observed differences are likely due to chance or if they're statistically significant.

Z-Score

A Z-score is a statistical measure that describes a value's relation to the mean of a group of values, measured in terms of standard deviations. In hypothesis testing, it tells us how far from the mean or expected value our observed proportion is, allowing us to assess the significance of our results.
For the mannequin study, the Z-score helps compare the difference in sample proportions to the standard error, calculated as:

• \( Z = \frac{p_1 - p_2}{SE} \)

Where \( p_1 \) and \( p_2 \) are the sample proportions, and \( SE \) is the standard error.
The Z-score can tell us how many standard deviations away our observed difference is from the mean. A higher Z-score indicates a more significant difference, leading us closer to concluding that the variable is impactful in the study context.

P-Value

The p-value is a probability metric used in hypothesis testing to determine the evidence against a null hypothesis. It helps us understand if our findings are due to random chance or if they can indeed be considered significant.
In our scenario, we calculate the p-value by comparing the Z-score with values from a standard normal distribution. This comparison helps evaluate how likely it is for the observed difference to occur if the null hypothesis were true.
Generally:

• If the p-value is less than a chosen significance level (often 0.05), it suggests there is strong evidence against the null hypothesis, indicating that the observed difference is significant.
• If the p-value is higher, it suggests that any observed difference is likely due to random variability in the sample.

The decision to use mannequins with or without heads hinges on these insights, as the p-value informs us whether the difference in buying intentions is truly significant.