The mean per capita consumption of milk per year is 153 liters with a standard deviation of 27 liters.
If a sample of 90 people is randomly selected, what is the probability that the sample mean would be less than 146.97 liters? Round your answer to four decimal places.
Answer
How to enter your answer (opens in new window)

Step 1 ：We are given a problem of probability involving normal distribution. The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution. This theorem applies here as we have a large sample size of 90 people.

Step 2 ：We are given the population mean \(\mu = 153\) liters, the population standard deviation \(\sigma = 27\) liters, and the sample size \(n = 90\) people. We are asked to find the probability that the sample mean \(\bar{x}\) is less than 146.97 liters.

Step 3 ：To solve this, we can use the formula for the z-score, which is \(\frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\). The z-score tells us how many standard deviations an element is from the mean. We can then use the z-score to find the probability.

Step 4 ：Substituting the given values into the z-score formula, we get \(z = -2.1187260323128143\).

Step 5 ：Using the z-score, we can find the probability that the sample mean is less than 146.97 liters. The probability is \(p = 0.0171\).

Step 6 ：Final Answer: The probability that the sample mean would be less than 146.97 liters is \(\boxed{0.0171}\).