Problem 10 - Basic Statistics for Business and Economics with Student

Question

The Roper Organization conducted identical surveys 5 years apart. One question asked of women was "Are most men basically kind, gentle, and thoughtful?" The earlier survey revealed that, of the 3,000 women surveyed, 2,010 said that they were. The later revealed 1,530 of the 3,000 women surveyed thought that men were kind, gentle, and thoughtful. At the .05 level, can we conclude that women think men are less kind, gentle, and thoughtful in the later survey compared with the earlier one?

Step-by-Step Solution

Answer

Women believe men are less kind in the later survey.

Step 1: State the Hypotheses

First, we need to identify the null and alternative hypotheses. The null hypothesis (\( H_0 \)) states that there is no change in women's opinions about men between the two surveys. In other words, the proportion of women who think men are kind is the same in both surveys. The alternative hypothesis (\( H_1 \)) states that fewer women think men are kind in the later survey.\[H_0: p_1 \leq p_2\]\[H_1: p_1 > p_2\]where \( p_1 \) is the proportion of women in the earlier survey and \( p_2 \) in the later survey, who think men are kind.

Step 2: Calculate Proportions

Next, calculate the proportions of women in both surveys who thought men were kind, gentle, and thoughtful.Earlier survey: \[p_1 = \frac{2010}{3000} = 0.67\]Later survey:\[p_2 = \frac{1530}{3000} = 0.51\]

Step 3: Calculate the Standard Error

Compute the standard error for the difference between the two proportions. The standard error \( SE \) is given by the formula:\[SE = \sqrt{\frac{p_1 (1-p_1)}{n_1} + \frac{p_2 (1-p_2)}{n_2}}\]Where \( n_1 \) and \( n_2 \) are the sample sizes for each survey.\[SE = \sqrt{\frac{0.67(1-0.67)}{3000} + \frac{0.51(1-0.51)}{3000}} = 0.0129\]

Step 4: Calculate the Test Statistic

Now, calculate the test statistic (\( Z \)) for the difference between the two proportions:\[Z = \frac{p_1 - p_2}{SE} = \frac{0.67 - 0.51}{0.0129} \approx 12.4\]

Step 5: Determine Critical Value and Conclusion

We are using a significance level of \( \alpha = 0.05 \). Since this is a one-tailed test, we compare the \( Z \)-value with the critical value for a one-tailed test at \( 0.05 \) which is approximately \( 1.645 \).Since \( 12.4 \) is greater than \( 1.645 \), we reject the null hypothesis.This means that there is sufficient evidence to conclude that women think men are less kind, gentle, and thoughtful in the later survey compared to the earlier one.

Key Concepts

Proportion Comparison

Proportion comparison is an essential concept when analyzing surveys or experiments in statistics. Here, we look at the ratio of a subgroup within the entire group to determine if there are differences across various samples. In our exercise, we're comparing the proportion of women who perceive men as kind, gentle, and thoughtful in two separate surveys conducted 5 years apart.
To do this, we calculate the proportion of respondents with positive responses in each survey:

• In the first survey, 2,010 out of 3,000 women, resulting in a proportion of 0.67.
• In the later survey, 1,530 out of 3,000 women, yielding a proportion of 0.51.

This difference in proportions reflects a change in perception over time, which can be formally checked using hypothesis testing. Identifying these proportions allows us to see that the later survey shows a decline in the positive perception of men. This step sets the stage for further analysis through sophisticated statistical tests.

Z-Test

A Z-test is a statistical method used to determine if there's a significant difference between two population proportions. For our exercise, the Z-test compares whether the decrease in the proportion of women opining positively about men in the later survey is statistically significant.
The Z-test involves calculating the difference between the two proportions and dividing it by the standard error of the difference between these proportions. This gives us the Z-score, which tells us how many standard deviations away the difference is from the mean, essentially showing if the difference is large enough to be statistically significant.
In the exercise, after computing the Z-score, we compared it to a critical value from a Z-table; for a one-tailed test at a 0.05 significance level, this value is 1.645. Our calculated Z-value, 12.4, is significantly greater than 1.645, providing strong evidence against the null hypothesis and suggesting a real change in women's perceptions. The Z-test, therefore, plays a crucial role in determining whether observed differences are meaningful or could have occurred by random chance.

Standard Error Calculation

Standard error (SE) is a measure used to quantify the amount of variation or dispersion of sample proportion estimates from the actual population proportions. Calculating the SE helps in assessing the precision of the sample estimate and forming the basis for tests like the Z-test.
In our context, the SE for the difference between the two proportions is calculated using a specific formula: \[ SE = \sqrt{ \frac{p_1 (1-p_1)}{n_1} + \frac{p_2 (1-p_2)}{n_2} } \]where \( p_1 \) and \( p_2 \) are the sample proportions from the first and later surveys, and \( n_1 \) and \( n_2 \) are the sample sizes. When we substitute the values from the exercise, we get an SE of 0.0129.
This small standard error indicates that the differences observed in proportions are measured with some precision. The SE helps transform the observed differences into a standard form, allowing for the application of the Z-test to assess significance. Proper calculation of SE underpins the accuracy and reliability of the statistical analysis performed.