To solve this problem, I need specific information from "Illustration 3", which is not provided in the current image. Specifically, I need: The population mean* ($\mu$) The population standard deviation* ($\sigma$) The sample size* ($n$) for each sample The total number of samples* ($N_{total}$) being considered. Assuming these values were available, here are the steps to solve the problem using the Central Limit Theorem: Step 1: Calculate the standard error of the mean. The standard error of the mean ($\sigma_{\bar{x}}$) is calculated as: $$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$ Step 2: Standardize the sample means to z-scores. For a given sample mean $\bar{x}$, the z-score is calculated as: $$ z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} $$ Step 3: Calculate the probabilities using the standard normal distribution. a) For the mean between 66.8 and 68.3 in: 1. Calculate the z-score for $\bar{x}_1 = 66.8$: $$ z_1 = \frac{66.8 - \mu}{\sigma_{\bar{x}}} $$ 2. Calculate the z-score for $\bar{x}_2 = 68.3$: $$ z_2 = \frac{68.3 - \mu}{\sigma_{\bar{x}}} $$ 3. Find the probability $P(66.8 < \bar{x} < 68.3)$, which is equivalent to $P(z_1 < Z < z_2)$. This can be found using a standard normal distribution table or calculator: $$ P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1) $$ b) For the mean less than 66.4 in: 1. Calculate the z-score for $\bar{x}_3 = 66.4$: $$ z_3 = \frac{66.4 - \mu}{\sigma_{\bar{x}}} $$ 2. Find the probability $P(\bar{x} < 66.4)$, which is equivalent to $P(Z < z_3)$. This can be found using a standard normal distribution table or calculator. Step 4: Calculate the expected number of samples. Multiply the probabilities found in Step 3 by the total number of samples ($N_{total}$). a) Expected number of samples between 66.8 and 68.3 in: $$ N_a = P(66.8 < \bar{x} < 68.3) \times N_{total} $$ b) Expected number of samples less than 66.4 in: $$ N_b = P(\bar{x} < 66.4) \times N_{total} $$ Please provide the details from "Illustration 3" (population mean, population standard deviation, sample size, and total number of samples) to get the numerical answers. That's 2 down. 3 left today — send the next one.