Problem 37 - An Introduction to Mathematical Statistics and Its Applications

Question

A total of 8605 students are enrolled full-time at State University this semester, 4134 of whom are women. Of the 6001 students who live on campus, 2915 are women. Can it be argued that the difference in the proportion of men and women living on campus is statistically significant? Carry out an appropriate analysis. Let \(\alpha=0.05\).

Step-by-Step Solution

Answer

To sum up, to solve this problem, we first need to understand the data presented and set up the hypothesis. We then calculate the proportions and the test statistic. If the P-value we get from this statistic is smaller than our significance level, we would reject our null hypothesis and assert that there is indeed a significant difference in the proportions.

Step 1: Understand the Data

Start by getting a clear understanding of the data provided. Here, the total number students enrolled full-time is 8605, out of which 4134 are women. Of the 6001 students who live on campus, 2915 are women.

Step 2: Set Up the Hypothesis

The null hypothesis (H0) assumes that there is no difference in the proportion of men and women living on campus: \(p1 = p2\). The alternative hypothesis (H1) assumes there is a difference in the proportion of men and women living on campus: \(p1 \neq p2\). Here, \(p1\) is the proportion of women who live on-campus and \(p2\) is the proportion of total students who live on-campus.

Step 3: Calculate the Proportions

We can express the proportion of women living on campus as \(p1 = \frac{2915}{6001}\), and the proportion of all full-time students living on campus as \(p2 = \frac{6001}{8605}\).

Step 4: Calculate the Test Statistic

The test statistic (z) for comparing two proportions is calculated using the formula: \(z = \frac{{(p1-p2) - 0}}{{\sqrt{\frac{{{p*(1-p)}}{n1}} + \frac{{{p*(1-p)}}{n2}}}}}\), where \(p = \frac{(x1+x2)}{(n1+n2)}\) is the combined sample proportion, \(x1\) and \(x2\) are the number of successes in each sample and \(n1\) and \(n2\) are the sizes of two samples.

Step 5: Determine the P-Value

Calculate the P-value using the normal distribution table for the calculated z-score.

Step 6: Decision

If the P-value is less than the given significance level (\(\alpha = 0.05\)), we reject the null hypothesis.

Key Concepts

Understanding Statistical Significance

Statistical significance is at the heart of hypothesis testing. It helps us decide if the results we observe are meaningful or could just be due to random chance. Imagine we have a coin, and we want to know if it's fair. If we flip it 100 times, and it lands on heads 60 times, is that unusual? This is where statistical significance comes in. We use it to figure out if the observed outcome, like flipping 60 heads out of 100, is likely to happen randomly if the coin is indeed fair. In hypothesis testing, a significance level (\(\alpha\)) is predefined, often chosen as 0.05. It represents a 5% risk that we might falsely claim there's an effect or difference when there isn't one. By establishing this threshold, we can determine the likelihood that our data reflects something true about the population.

• If our test results show a probability (P-value) lower than this threshold, it implies statistical significance.
• This would mean our observed results are less likely to occur randomly, so we could dismiss the null hypothesis.

Knowing the significance level helps us to make more informed decisions about our data's reliability.

Proportion Testing Insights

Proportion testing is a statistical method that helps compare two proportions. For instance, we might want to know if there's a real difference between the proportion of men and women living on campus. To find this out, we utilize proportion testing as part of our hypothesis testing process. Let's break down the process. First, we calculate each group's proportion. In our example:

• The proportion of women living on campus is calculated as \(p1 = \frac{2915}{6001}\)
• The overall proportion of full-time students living on campus is \(p2 = \frac{6001}{8605}\).

By comparing these two proportions, we can evaluate if the differences we see are significant or just happen by random variation. Proportion testing often depends on computing a test statistic, such as a z-score, to summarize the data's difference relative to its variability. This test statistic plays a crucial role in determining statistical significance, which helps us understand whether the difference in proportions is noteworthy or could be by mere chance.

The Null Hypothesis Explained

The null hypothesis (\(H_0\)) is a foundational concept in statistical analysis. It represents an initial assumption that there is no effect or difference in the context we're studying. Think of it as a statement of no change or status quo. In our scenario, the null hypothesis asserts no difference in the proportions of men and women living on campus.When we perform hypothesis testing, our main goal is to assess whether there's enough evidence to reject this null hypothesis. This doesn't prove the alternative directly but indicates that the difference is significant enough not to be random.Remember:

• The null hypothesis is always paired with the alternative hypothesis.
• If our testing results show that the null hypothesis is unlikely to be true, we reject it in favor of the alternative.

The process of hypothesis testing revolves around challenging the null hypothesis, using data to make a decision with a quantified level of confidence.

Understanding the Alternative Hypothesis

The alternative hypothesis (\(H_1\) or sometimes \(H_a\)) offers the contrast against the null hypothesis. It suggests that there's a real effect, difference, or relationship in the context of the study. In our case, the alternative hypothesis claims there's a difference in the proportion of men and women living on campus. This is what we hope to provide evidence for through our statistical test.When conducting hypothesis testing:

• The alternative hypothesis is what researchers aim to support.
• It's often formulated such that it contrasts with the null hypothesis.

The alternative hypothesis isn't directly proven; instead, rejection of the null hypothesis increases the likelihood that the alternative is true. Choosing between the null and alternative hypotheses depends on the calculated P-value and the pre-set significance level, helping us make informed decisions on the investigative queries at hand.