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Chapter 22: Problem 838

The marketing research firm of Burrows. Heller and Larimer wants to estimate the proportions of men and women who are familiar with a shoe polish. In a sample (random) of 100 men and 200 women it is found that 20 men and 60 women had used this particular shoe polish. Compute a \(95 \%\) confidence Interval for the difference in proportions between men and women familiar with the product. Use this to test the hypothesis that the proportions are equal.

Short Answer

Expert verified
Our 95% confidence interval for the difference in proportions between men and women familiar with the shoe polish is approximately (-0.1009, 0.0009). Since this interval contains 0, we cannot reject the null hypothesis and therefore, there is no significant evidence to suggest that the proportions of men and women familiar with the shoe polish are different.

Step by step solution

01

Compute the sample proportions

First, let's calculate the sample proportions for men and women. We use the formula: \( p_{men} = \frac{number\ of\ men\ who\ used\ the\ shoe\ polish}{total\ number\ of\ men\ in\ the\ sample} \) \( p_{women} = \frac{number\ of\ women\ who\ used\ the\ shoe\ polish}{total\ number\ of\ women\ in\ the\ sample} \) Using the given data: \( p_{men} = \frac{20}{100} = 0.2 \) \( p_{women} = \frac{60}{200} = 0.3 \)
02

Compute the standard error

Now, let's calculate the standard error of the difference in proportions using the formula: \( SE = \sqrt{ \frac{p_{men}(1- p_{men})}{n_{men}} + \frac{p_{women}(1- p_{women})}{n_{women}}} \) Substituting the values: \( SE = \sqrt{ \frac{0.2(1- 0.2)}{100} + \frac{0.3(1- 0.3)}{200}} = \sqrt{0.0016 + 0.00105} = \sqrt{0.00265} \approx 0.0515 \)
03

Calculate the confidence interval

We want to compute a 95% confidence interval for the difference in proportions. This means that our critical z-score (z-value) is 1.96. The confidence interval is given by: \( CI = (p_{men} - p_{women}) \pm z \cdot SE \) Let's calculate the lower and upper bounds of the confidence interval: Lower bound: \( (0.2 - 0.3) - 1.96 \cdot 0.0515 \approx -0.1009 \) Upper bound: \( (0.2 - 0.3) + 1.96 \cdot 0.0515 \approx 0.0009 \) So, the 95% confidence interval for the difference in proportions between men and women is approximately (-0.1009, 0.0009).
04

Test the hypothesis

Lastly, we need to test the hypothesis that the proportions of men and women familiar with the shoe polish are equal. That is: H0: \( p_{men} - p_{women} = 0 \) Ha: \( p_{men} - p_{women} \neq 0 \) Since the confidence interval (-0.1009, 0.0009) contains 0, we cannot reject the null hypothesis. Thus, we conclude that there's no significant evidence to suggest that the proportions of men and women familiar with the shoe polish are different.

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Key Concepts

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

Difference in Proportions
When comparing two groups, such as men and women in this exercise, we often look at the difference in proportions to see if the use varies significantly between them. Here, we're evaluating how many from each group are familiar with a shoe polish. This is calculated as the difference between the sample proportions of men and women.

The formula for the difference in proportions is:
  • For men: \( p_{men} = \frac{20}{100} = 0.2 \)
  • For women: \( p_{women} = \frac{60}{200} = 0.3 \)
Thus, the difference \((p_{men} - p_{women})\) is \(0.2 - 0.3 = -0.1\). This means that a smaller proportion of men are familiar with the shoe polish compared to women.
Hypothesis Testing
Hypothesis testing helps determine whether there is enough statistical evidence in a sample to infer that a certain condition holds for the entire population. In our example, we want to test if the difference in proportions is significantly different from zero.

Here's how we set it up:
  • Null Hypothesis \( (H_0) \): The proportions are equal, \( p_{men} - p_{women} = 0 \).
  • Alternative Hypothesis \( (H_a) \): The proportions are not equal, \( p_{men} - p_{women} eq 0 \).
We use the confidence interval to decide whether to accept or reject the null hypothesis. If the interval includes 0, as in our case (-0.1009, 0.0009), we don't have enough evidence to reject the null hypothesis. This suggests that there is no significant difference in the familiarity with the product between the two groups.
Standard Error
Standard Error (SE) measures the variation of a sample statistic. It tells us how much the proportion we compute from the sample is likely to vary from the true proportion in the population.

To compute SE for the difference in proportions, we use the following formula:
  • \( SE = \sqrt{ \frac{p_{men}(1- p_{men})}{n_{men}} + \frac{p_{women}(1- p_{women})}{n_{women}}} \)
Plugging in our values, we calculate it as approximately 0.0515.

This SE helps us build a confidence interval around the sample proportion difference. If this interval is wide, it indicates more variability and less certainty in our estimate.

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Most popular questions from this chapter

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