Questions: The average length of "short hospital stays" for men is slightly longer than that for women, 5.6 days versus 4.7 days. A random sample of recent hospital stays for both men and women revealed the following. The goal of the study is to determine if there is sufficient evidence to conclude, at α=0.1, that the average hospital stay for men is longer than the average hospital stay for women. Men Women Sample size 38 33 Sample mean 5.6 days 4.7 days Population standard deviation 1.3 days 1.7 days What would be the critical value(s).

The average length of "short hospital stays" for men is slightly longer than that for women, 5.6 days versus 4.7 days. A random sample of recent hospital stays for both men and women revealed the following. The goal of the study is to determine if there is sufficient evidence to conclude, at α=0.1, that the average hospital stay for men is longer than the average hospital stay for women.

Men Women
Sample size 38 33
Sample mean 5.6 days 4.7 days
Population standard deviation 1.3 days 1.7 days

What would be the critical value(s).

Transcript text: The average length of "short hospital stays" for men is slightly longer than that for women, 5.6 days versus 4.7 days. A random sample of recent hospital stays for both men and women revealed the following. The goal of the study is to determine if there is sufficient evidence to conclude, at $\alpha=0.1$, that the average hospital stay for men is longer than the average hospital stay for women. \begin{tabular}{|l|c|c|} \hline & Men & Women \\ \hline Sample size & 38 & 33 \\ \hline Sample mean & 5.6 days & 4.7 days \\ \hline Population standard deviation & 1.3 days & 1.7 days \\ \hline \end{tabular} What would be the critical value(s).

Solution

Solution Steps

To determine if there is sufficient evidence to conclude that the average hospital stay for men is longer than the average hospital stay for women, we can perform a hypothesis test for the difference between two means. Given the significance level $\alpha = 0.1$, we need to find the critical value for a one-tailed test.

Solution Approach

Identify the null and alternative hypotheses:
- Null hypothesis ($H_0$): $\mu_{men} \leq \mu_{women}$
- Alternative hypothesis ($H_1$): $\mu_{men} > \mu_{women}$
Determine the significance level $\alpha = 0.1$.
Since this is a one-tailed test, find the critical value corresponding to $\alpha = 0.1$ from the standard normal distribution (Z-distribution).

Step 1: State the Hypotheses

We start by stating the null and alternative hypotheses:

Null hypothesis ($H_0$): $\mu_{\text{men}} \leq \mu_{\text{women}}$
Alternative hypothesis ($H_1$): $\mu_{\text{men}} > \mu_{\text{women}}$

Step 2: Determine the Significance Level

The significance level is given as $\alpha = 0.1$.

Step 3: Find the Critical Value

For a one-tailed test at $\alpha = 0.1$, we need to find the critical value from the standard normal distribution (Z-distribution). The critical value corresponding to $\alpha = 0.1$ is approximately $1.2816$.