SOLUTION: A random variable is normally distributed. It has a mean of 245 and a standard deviation of 21. a.) If you take a sample of size 10, can you say what the shape of the distribut

Question 1186329: A random variable is normally distributed. It has a mean of 245 and a standard deviation of 21.
a.) If you take a sample of size 10, can you say what the shape of the distribution for the sample mean is? Why?
b.) For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean.
c.) For a sample of size 10, find the probability that the sample mean is more than 241.
d.) If you take a sample of size 35, can you say what the shape of the distribution of the sample mean is? Why?
e.) For a sample of size 35, state the mean of the sample mean and the standard deviation of the sample mean.
f.) For a sample of size 35, find the probability that the sample mean is more than 241.
g.) Compare your answers in part c and f. Why is one smaller than the other?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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.

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