A population of values has a normal distribution with $\mu=180.5$ and $\sigma=27.8$. If a random sample of size $n=22$ is selected,
b. Find the probability that a sample of size $n=22$ is randomly selected with a mean less than 162.7 . Round your answer to four decimals.
\[
P(M< 162.7)=
\]
Answer
Thus, the probability that a sample of size \(n=22\) is randomly selected with a mean less than 162.7 is approximately \(\boxed{0.0013}\).
Step 1 ：We are given a population of values with a normal distribution where the mean \(\mu\) is 180.5 and the standard deviation \(\sigma\) is 27.8. A random sample of size \(n=22\) is selected.
Step 2 ：We are asked to find the probability that the sample mean is less than 162.7. This can be represented as \(P(M<162.7)\).
Step 3 ：We can use the Central Limit Theorem to solve this problem. The Central Limit Theorem states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. This approximation is generally good if the sample size is at least 30. However, for populations that are approximately normally distributed, the approximation can be good even if the sample size is less than 30.
Step 4 ：The mean of the sample means is the same as the population mean, and the standard deviation of the sample means (also known as the standard error) is the population standard deviation divided by the square root of the sample size.
Step 5 ：We can standardize the sample mean by subtracting the population mean and dividing by the standard error. This gives us a z-score, which we can look up in a standard normal distribution table to find the probability.
Step 6 ：Given the values, we have \(\mu = 180.5\), \(\sigma = 27.8\), \(n = 22\), and \(\bar{x} = 162.7\).
Step 7 ：We calculate the standard error (se) as \(\sigma / \sqrt{n}\), which gives us approximately 5.93.
Step 8 ：We then calculate the z-score as \((\bar{x} - \mu) / se\), which gives us approximately -3.00.
Step 9 ：Looking up the z-score in a standard normal distribution table, we find that the probability is approximately 0.0013.
Step 10 ：Thus, the probability that a sample of size \(n=22\) is randomly selected with a mean less than 162.7 is approximately \(\boxed{0.0013}\).
