HW E8: Central Limit Theorem Progres Score: $12 / 16 \quad 6 / 8$ answered Question 7 A population of values has a normal distribution with $\mu=94.8$ and $\sigma=16$. You intend to draw a random sample of size $n=218$. Find the probability that a single randomly selected value is less than 91.8 . \[ P(X<91.8)= \] Find the probability that a sample of size $n=218$ is randomly selected with a mean less than 91.8. \[ P(M<91.8)= \] Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or zscores rounded to 3 decimal places are accepted. Question Help: D Post to forum Submit Question

Step 1 ：Calculate the z-score for a single randomly selected value less than 91.8 using the formula \(Z = \frac{X - \mu}{\sigma}\). Here, \(X = 91.8\), \(\mu = 94.8\), and \(\sigma = 16\).

Step 2 ：Substitute the values into the formula to get \(Z = \frac{91.8 - 94.8}{16} = -0.1875\).

Step 3 ：Look up the z-score in a standard normal distribution table or use a calculator to find the probability. The probability that Z is less than -0.1875 is approximately 0.4255.

Step 4 ：Calculate the standard error for a sample of size n=218 using the formula \(\sigma/\sqrt{n}\). Here, \(\sigma = 16\) and \(n = 218\).

Step 5 ：Substitute the values into the formula to get \(\sigma/\sqrt{n} = 16/\sqrt{218} \approx 1.086\).

Step 6 ：Calculate the z-score for a sample mean of 91.8 using the formula \(Z = \frac{X - \mu}{\sigma}\). Here, \(X = 91.8\), \(\mu = 94.8\), and \(\sigma = 1.086\).

Step 7 ：Substitute the values into the formula to get \(Z = \frac{91.8 - 94.8}{1.086} = -2.76\).

Step 8 ：Look up the z-score in a standard normal distribution table or use a calculator to find the probability. The probability that Z is less than -2.76 is approximately 0.0029.

Step 9 ：So, the answers are \(\boxed{P(X<91.8) = 0.4255}\) and \(\boxed{P(M<91.8) = 0.0029}\).