4. [0/3 Points] DETAILS MY NOTES A random sample of recent hospital stays for 32 men and 30 women revealed the following. At 0.01 level, is there sufficient evidence to conclude that the average hospital stay for men is longer than the average hospital stay for women? What is the Critical Value (If there are two, separate them with a comma) $$ = $$

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To determine the critical value for this hypothesis test, we need to follow these steps:

1. **Identify the Hypotheses**:
   - Null Hypothesis ($$H_0$$): The average hospital stay for men is less than or equal to that for women ($$\mu_m \leq \mu_w$$).
   - Alternative Hypothesis ($$H_a$$): The average hospital stay for men is greater than that for women ($$\mu_m > \mu_w$$).

2. **Determine the Significance Level**:
   - The significance level ($$\alpha$$) is given as 0.01.

3. **Determine the Type of Test**:
   - Since we are testing if the average hospital stay for men is greater than that for women, this is a one-tailed test.

4. **Find the Critical Value**:
   - For a one-tailed test at the 0.01 significance level, we need to find the critical value from the standard normal distribution (Z-distribution) or t-distribution, depending on whether we know the population standard deviations or not.

Since we have a sample size of 32 men and 30 women, we can use the t-distribution because the sample sizes are relatively small and we likely do not know the population standard deviations.

- The degrees of freedom (df) can be calculated using the formula for two independent samples:
  $$
  df = n_1 + n_2 - 2 = 32 + 30 - 2 = 60
  $$

Using a t-table or calculator for a one-tailed test at $$\alpha = 0.01$$ with 60 degrees of freedom, we find the critical t-value.

The critical t-value for $$df = 60$$ at $$\alpha = 0.01$$ (one-tailed) is approximately **2.390**.

Thus, the critical value is:
$$
\text{Critical Value} = 2.390
$$

If you are using a Z-distribution instead (which is less common for small sample sizes), the critical value would be approximately **2.33**.

In summary, the critical value for this test is approximately **2.390** (using t-distribution).
The critical value for the test is approximately 2.390 (using t-distribution).

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