1. **Problem Statement:** We have a population with mean $\mu=140$ and standard deviation $\sigma=30$. We take a random sample of size $n=75$. We want to find probabilities related to the sample mean $\bar{X}$.\n\n2. **Formula and Explanation:** The sampling distribution of the sample mean $\bar{X}$ has mean $\mu_{\bar{X}}=\mu=140$ and standard deviation (standard error) $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{30}{\sqrt{75}}$.\n\n3. Calculate standard error: $$\sigma_{\bar{X}}=\frac{30}{\sqrt{75}}=\frac{30}{8.6603}=3.4641.$$\n\n4. We use the normal distribution to approximate probabilities for $\bar{X}$ because of the Central Limit Theorem, even though the population is skewed, the sample size is large ($n=75$).\n\n**a. Probability that $\bar{X}$ is between 132 and 136:**\n- Convert to $Z$-scores: $$Z=\frac{\bar{X}-\mu}{\sigma_{\bar{X}}}.$$\n- For 132: $$Z_1=\frac{132-140}{3.4641}=-2.309.$$\n- For 136: $$Z_2=\frac{136-140}{3.4641}=-1.155.$$\n- Find $P(132<\bar{X}<136)=P(Z_1