a) Does the CLT hold? State condition why. b) What is the probability that the average scores are more than 70? c) What score describes a student’s rank to be at the top 5% of the... a) Does the CLT hold? State condition why. b) What is the probability that the average scores are more than 70? c) What score describes a student’s rank to be at the top 5% of the midterm scores? Read more

Steps to Solve

1. Checking the Central Limit Theorem (CLT)

For the CLT to hold, the sample size should be large enough or the population should be normally distributed. Since the sample size is 36 (which is greater than 30), we can say that the CLT applies regardless of the population distribution because the sample size is considered large.

2. Calculating the probability of average scores greater than 70

We will use the z-score formula to find the probability.

First, calculate the mean and standard deviation of the sample mean:

• Mean ($\mu$) = 73
• Standard deviation of the sample mean ($\sigma_{\bar{x}}$) = $\frac{\sigma}{\sqrt{n}} = \frac{7.3}{\sqrt{36}} = \frac{7.3}{6} \approx 1.2167$

Next, find the z-score for an average score of 70:

$$ z = \frac{X - \mu}{\sigma_{\bar{x}}} = \frac{70 - 73}{1.2167} \approx -2.464 $$

Using the z-table or a calculator, find the probability corresponding to this z-score.

3. Finding the probability using a z-table or calculator

Look up the z-score of -2.464 in the z-table. It corresponds to a probability of about 0.0069. Therefore, the probability that the average scores are more than 70 is:

$$ P(X > 70) = 1 - P(Z < -2.464) \approx 1 - 0.0069 \approx 0.9931 $$

4. Finding the score for the top 5%

To find the score that corresponds to the top 5%, we need to find the z-score that corresponds to 95% in the z-table, which is approximately 1.645.

Using the formula for the score:

$$ X = \mu + z \cdot \sigma_{\bar{x}} $$

Substituting values:

$$ X = 73 + 1.645 \cdot 1.2167 \approx 73 + 2.000 \approx 75 $$