Question

The mean per capita consumption of milk per year is 132 liters with a variance of 625. If a sample of 196 people is randomly selected, what is the probability that the sample mean would differ from the true mean by more than 5.15 liters?

Answer 1

To calculate the probability that the sample mean of milk consumption differs from the true mean by more than 5.15 liters, use the Central Limit Theorem to standardize the sample mean, find the z-scores for a 5.15-liter difference, and then use the z-table to find the probabilities for the upper and lower tails.

Step-by-step explanation:

To determine the probability that the sample mean would differ from the true mean by more than 5.15 liters when the mean per capita consumption of milk per year is 132 liters with a variance of 625, we will use the Central Limit Theorem (CLT). The CLT tells us that the sampling distribution of the sample mean will be approximately normally distributed when the sample size is large enough (in this case, n=196 is sufficiently large).

To find this probability, we need to standardize the sample mean using the following formula for the z-score:

1. Calculate the standard error (SE) of the mean, which is the standard deviation of the sampling distribution of the sample mean. SE = sqrt(variance / sample size) = sqrt(625 / 196).
2. Find the z-score for the difference of 5.15 liters from the true mean. Z = (X - mean) / SE.
3. Use the standard normal distribution table to find the probability corresponding to the calculated z-scores for both the upper and lower tails, as this is a two-tailed test.
   

Since the exact calculations and z-table references are not provided here, you would perform this calculation to find the z-scores and then use the standard normal distribution to find the probabilities for each tail. Lastly, add the probabilities of both tails to find the total probability that the sample mean will differ from the true mean by more than 5.15 liters.