Question 1185821
Here's how to address each part of the hypothesis test:

**(a) Two Means or Two Proportions?**

This is a test of *two means* because we are comparing the average number of math classes taken by graduates from two different colleges.

**(b) Known or Unknown Population Standard Deviations?**

The population standard deviations are *unknown*. We are given the *sample* standard deviations.

**(c) Which Distribution to Use?**

Since the population standard deviations are unknown and the sample sizes are small (n1 = 11, n2 = 9), we use the *t-distribution*.  Because the sample sizes are small, and the population standard deviations are unknown, we must assume that the population variances are equal.

**(d) What is the Random Variable?**

The random variable is the *difference* between the two sample means:  x̄1 - x̄2, where x̄1 is the mean number of math classes for College A graduates and x̄2 is the mean number of math classes for College B graduates.

**(e) Null and Alternative Hypotheses:**

*   **Null Hypothesis (H0):** The average number of math classes taken by graduates from College A is equal to or less than the average number of math classes taken by graduates from College B.  μ1 ≤ μ2

*   **Alternative Hypothesis (H1):** The average number of math classes taken by graduates from College A is greater than the average number of math classes taken by graduates from College B. μ1 > μ2  (This is what the community group believes.)

**(f) Right, Left, or Two-Tailed Test?**

This is a *right-tailed* test because the alternative hypothesis states that the mean for College A is *greater* than the mean for College B.

**(g) What is the p-value?**

1.  **Calculate the test statistic:**
    Because we are assuming the population variances are equal, we use the following formula for the t-statistic:
    ```
    t = (x̄1 - x̄2) / sqrt(s_p^2 * (1/n1 + 1/n2))
    ```
    Where s_p is the pooled standard deviation, calculated as:
    ```
    s_p = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1 + n2 - 2))
    ```
    Plugging in the given values, we get s_p = sqrt(((10)*1.5^2 + (8)*1^2) / 18) = 1.32
    Then, the t statistic is calculated as:
    ```
    t = (4 - 3.5) / sqrt(1.32^2 * (1/11 + 1/9)) = 0.5/(1.32*0.447) = 0.84
    ```

2.  **Degrees of Freedom:** df = n1 + n2 - 2 = 11 + 9 - 2 = 18

3.  **Find the p-value:** Using a t-table or calculator with df = 18 and t = 0.84, we find the p-value.  Since it is a right-tailed test, the p-value is the area to the *right* of the calculated t-statistic.  The p-value is approximately 0.208.

**(h) Reject or Not Reject the Null Hypothesis?**

*   **Significance Level:** α = 0.01

*   **Decision:** Since the p-value (0.208) is *greater* than the significance level (0.01), we *fail to reject* the null hypothesis.

**(Conclusion):**

At a 1% significance level, there is *not* sufficient evidence to conclude that graduates from College A take more math classes on average than graduates from College B.