Problem 31 - Basic Statistics for Business and Economics with Student

Question

In a poll recently conducted at Iowa State University, 68 out of 98 male students and 45 out of 85 female students expressed "at least some support" for implementing an "exit strategy" from Iraq. Test at the .05 significance level the null hypothesis that the population proportions are equal against the two-tailed alternative.

Step-by-Step Solution

Answer

The null hypothesis is rejected; there is a significant difference between male and female proportions.

Step 1: Define Hypotheses

First, we define the null and alternative hypotheses. The null hypothesis (H_0) is that the population proportions of male and female students who support the exit strategy are equal. The alternative hypothesis (H_a) is that the population proportions are not equal. Formally, \[H_0: p_1 = p_2\]\[H_a: p_1 eq p_2\] where \(p_1\) is the proportion of male students supporting the strategy, and \(p_2\) is the proportion of female students.

Step 2: Calculate Sample Proportions

Calculate the sample proportions. For males: \[ \hat{p}_1 = \frac{68}{98} \approx 0.6939 \] For females: \[ \hat{p}_2 = \frac{45}{85} \approx 0.5294 \]

Step 3: Determine Pooled Proportion

Calculate the pooled proportion. The pooled proportion (\( \hat{p} \)) is\[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{68 + 45}{98 + 85} = \frac{113}{183} \approx 0.6175 \] where \(x_1 = 68\), \(x_2 = 45\), \(n_1 = 98\), \(n_2 = 85\).

Step 4: Calculate Standard Error

Determine the standard error (SE) using the formula: \[ SE = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \] Substituting the values: \[ SE = \sqrt{0.6175(1 - 0.6175)\left(\frac{1}{98} + \frac{1}{85}\right)} \approx 0.0725 \]

Step 5: Compute Z-test Statistic

Use the Z-test formula: \[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \] Substituting the values: \[ Z = \frac{0.6939 - 0.5294}{0.0725} \approx 2.268 \]

Step 6: Determine Critical Value and Decision

For a two-tailed test at the 0.05 significance level, the critical z-values are approximately ±1.96. Since our calculated Z-value (2.268) is greater than 1.96, we reject the null hypothesis.

Key Concepts

Understanding the Z-Test

In statistics, the Z-test is a powerful tool when you want to compare sample means or proportions. It's especially useful when you have a large sample size, and you know the population variance. The Z-test helps determine if the evidence you have from samples can be extended to the larger population.

In our specific case, the Z-test is used to check if there's a significant difference between the proportion of male and female students who support the exit strategy from Iraq.

• You first need to establish your null hypothesis (no difference in proportions) and alternative hypothesis (there is a difference).
• Calculate the standard error of the difference between sample proportions, which tells you how much variability you can expect by chance.
• Finally, you compute the Z-statistic, which tells you how far off your sample result is from the null hypothesis assumption.

A larger absolute Z-value indicates a stronger deviation from the null hypothesis, which can hint at a significant difference in population proportions.

Proportions Explained

Proportions are simply ratios that show the size of one piece compared to the whole. In our exercise, the proportion represents the part of the selected male or female students who support an exit strategy. This is calculated by dividing the number of supporters by the total number of respondents in each group.

Calculating sample proportions gives us a grounded idea of how large or small an observed sample attribute is in relation to its entire sample. Thus, it plays a crucial role in hypothesis testing:

• Sample proportions are compared to test if different groups show similar characteristics.
• If sample proportions differ a lot from one group to another, and this difference isn’t due to random variations, then they help in identifying statistically significant differences.

Understanding proportions aids in making data-driven decisions and recognizing patterns within data.

The Concept of Significance Level

The significance level, often denoted by α (alpha), is a threshold set by researchers before conducting a hypothesis test. It represents how willing we are to reject the null hypothesis when it's actually true, also known as the risk of a Type I error. In the given exercise, a 0.05 significance level means there's a 5% risk of wrongly concluding a difference exists when it actually doesn't.

To determine if an observed result is significant, we compare the computed Z-statistic with critical value thresholds:

• If a Z-statistic lies beyond the critical z-values, the result is considered statistically significant, rejecting the null hypothesis.
• For a 0.05 significance level in a two-tailed test, the critical z-values are approximately ±1.96.

Carefully setting and understanding significance levels are integral to retaining accuracy and integrity in statistical inference, helping us understand when a result is likely due to chance.