Question 1186329
Here's the solution:

**a.) Shape of the distribution for the sample mean (n=10):**

Yes. Because the original population is normally distributed, the sampling distribution of the sample mean will *also* be normally distributed, regardless of the sample size.

**b.) Mean and standard deviation of the sample mean (n=10):**

*   Mean of the sample mean (μₓ̄) = Population mean (μ) = 245
*   Standard deviation of the sample mean (σₓ̄) = Population standard deviation (σ) / √n = 21 / √10 ≈ 6.64

**c.) Probability that the sample mean is more than 241 (n=10):**

1.  **Calculate the z-score:**
    z = (x̄ - μ) / σₓ̄
    z = (241 - 245) / 6.64
    z ≈ -0.60

2.  **Find the probability:**
    Using a z-table or calculator, find the probability of z being *greater* than -0.60.
    P(z > -0.60) ≈ 0.7257

**d.) Shape of the distribution of the sample mean (n=35):**

Yes.  Again, since the original population is normally distributed, the sampling distribution of the sample mean will *also* be normally distributed, even with a larger sample size.

**e.) Mean and standard deviation of the sample mean (n=35):**

*   Mean of the sample mean (μₓ̄) = Population mean (μ) = 245
*   Standard deviation of the sample mean (σₓ̄) = Population standard deviation (σ) / √n = 21 / √35 ≈ 3.56

**f.) Probability that the sample mean is more than 241 (n=35):**

1.  **Calculate the z-score:**
    z = (x̄ - μ) / σₓ̄
    z = (241 - 245) / 3.56
    z ≈ -1.12

2.  **Find the probability:**
    Using a z-table or calculator, find the probability of z being *greater* than -1.12.
    P(z > -1.12) ≈ 0.8686

**g.) Comparison of probabilities and explanation:**

The probability in part f (n=35) is larger than the probability in part c (n=10). This is because the standard deviation of the sample mean is *smaller* for the larger sample size.  A smaller standard deviation means the sample means are more tightly clustered around the population mean.  Therefore, it's more likely that a sample mean from a larger sample will be closer to the population mean (and thus more likely to be above 241 in this case).  In simpler terms, larger samples provide more precise estimates.