91Ó°ÊÓ

Proportions in statistics help us to compare parts of a whole. They express how large one subgroup is compared to the entire group. In this context, proportions refer to the percentage of people who prefer computer-assisted instruction over traditional lectures.
The key formula for a proportion is \( p = \frac{x}{n} \), where \( x \) is the number of individuals indicating a choice (like preferring computer-assisted instruction) and \( n \) is the total number from the sample. Each sample here has 100 individuals: 83% of men and 75% of women preferred the computer method.
This setup involves comparing these two proportions to see if the preference signs are consistent. The proportions of interest are \( p_1 = 0.83 \) for men and \( p_2 = 0.75 \) for women. Analyzing these helps us understand if there’s a statistically significant difference in preferences between the two groups. If not, we conclude both men and women equally prefer computer-assisted instruction.

A confidence interval provides a range within which we expect the true difference between the proportions of two populations to lie, with a certain level of confidence. Here, we’ve calculated a 95% confidence interval for the difference \( p_1 - p_2 \), meaning we’re 95% confident that the true difference between men's and women’s preferences falls within this interval.
The formula used is:

• \( (p_1 - p_2) \pm z_{\alpha/2} \times SE \),

where \( SE \) stands for standard error and \( z_{\alpha/2} \) is the critical value for our confidence level, which is 1.96 for 95% confidence
Our earlier calculations showed that this interval is \([ -0.034, 0.194 ]\).
This interval includes zero, suggesting that the actual difference in proportions could be negative, positive, or zero. If zero is in the interval, it indicates no strong evidence of a proportional difference between men’s and women’s preferences.

In hypothesis testing, the critical value helps to determine the threshold for making decisions. It tells us how extreme a test statistic must be to reject the null hypothesis. At a significance level (\( \alpha = 0.05 \)), splitting the alpha into two tails for a two-tailed test gives us 0.025 in each tail of the distribution.
For one standard normal distribution, these critical values are approximately \( z = \pm 1.96 \). This means any test statistic beyond these boundaries would lead us to reject the null hypothesis. In this exercise, we used these values to compare against our test statistic.
Establishing critical values is crucial for maintaining the rigor of hypothesis tests, setting a clear, objective standard for decision making regarding the claim on proportion preferences.

The test statistic in hypothesis testing quantifies the difference between observed sample proportions and their expected values under the null hypothesis. It provides a metric for deciding whether this observed difference is significant enough to reject the null hypothesis.
To calculate it, first, we determine the pooled sample proportion:

• \( p = \frac{x_1 + x_2}{n_1 + n_2} = \frac{83 + 75}{200} = 0.79 \)

Next, we compute the standard error:

• \( SE = \sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.79 \times 0.21 \times \left(0.01 + 0.01 \right)} \approx 0.058 \)

Finally, the test statistic \( z \) is:

• \( z = \frac{p_1 - p_2}{SE} = \frac{0.83 - 0.75}{0.058} \approx 1.38 \)

Since this \( z \) value isn’t beyond our critical values of \( \pm 1.96 \), we fail to reject the null hypothesis, indicating no significant preference difference between genders.