Given a normal distribution with $\mu=100$ and $\sigma=20$, and given you select a sample of $n=16$, complete parts (a) through (d).
a. What is the probability that $\bar{X}$ is less than 90 ?
\[
\mathrm{P}(\overline{\mathrm{X}}< 90)=0.0228
\]
(Type an integer or decimal rounded to four decimal places as needed.)
b. What is the probability that $\bar{X}$ is between 90 and 90.5 ?
\[
\mathrm{P}(90< \overline{\mathrm{X}}< 90.5)=0.0059
\]
(Type an integer or decimal rounded to four decimal places as needed.)
c. What is the probability that $\bar{X}$ is above $101.8 ?$
\[
P(\bar{X}> 101.8)=0.3594
\]
(Type an integer or decimal rounded to four decimal places as needed.)
d. There is a $63 \%$ chance that $\bar{X}$ is above what value?
\[
\bar{x}=\square
\]
(Type an intêger or decimal rounded to two decimal places as needed.)

Step 1 ：Given a normal distribution with mean \(\mu = 100\) and standard deviation \(\sigma = 20\), and a sample size of \(n = 16\).

Step 2 ：For part (a), we need to find the probability that the sample mean is less than 90. This involves finding the z-score for 90 and then finding the corresponding probability from the standard normal distribution. The z-score is calculated as \(z = \frac{X - \mu}{\sigma/\sqrt{n}}\), which gives \(z = -2.0\). The corresponding probability is \(P = 0.0228\).

Step 3 ：For part (b), we need to find the probability that the sample mean is between 90 and 90.5. This involves finding the z-scores for 90 and 90.5 and then finding the corresponding probability from the standard normal distribution. The z-scores are calculated as \(z_1 = -2.0\) and \(z_2 = -1.9\). The corresponding probability is \(P = 0.0059\).

Step 4 ：For part (c), we need to find the probability that the sample mean is above 101.8. This involves finding the z-score for 101.8 and then finding the corresponding probability from the standard normal distribution. The z-score is calculated as \(z = 0.36\). The corresponding probability is \(P = 0.3594\).

Step 5 ：For part (d), we need to find the value of the sample mean that corresponds to a 63% chance of the sample mean being above that value. This involves finding the z-score that corresponds to a probability of 0.63 and then converting that z-score back to a value of the sample mean. The z-score is calculated as \(z = -0.33\). The corresponding sample mean is \(\bar{x} = 98.34\).

Step 6 ：Final Answer: a. The probability that \(\bar{X}\) is less than 90 is \(\boxed{0.0228}\). b. The probability that \(\bar{X}\) is between 90 and 90.5 is \(\boxed{0.0059}\). c. The probability that \(\bar{X}\) is above 101.8 is \(\boxed{0.3594}\). d. There is a 63% chance that \(\bar{X}\) is above \(\boxed{98.34}\).