Question

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 differ from the true mean by less than 5.11 liters?

Answer 1

The question pertains to using the Central Limit Theorem to find the probability that the sample mean of milk consumption for a sample of 90 people will not differ from the population mean by more than 5.11 liters.

Step-by-step explanation:

The student is asking about the probability that the sample mean of milk consumption from a sample of 90 people would differ from the population mean by less than 5.11 liters, given that the population mean is 153 liters and the standard deviation is 27 liters. To solve this, we use the Central Limit Theorem which states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the population's distribution shape, provided the sample size is large enough (typically n > 30).

We need to find the standard error (SE) of the mean, which is the standard deviation of the sampling distribution of the sample mean. The SE is calculated as the population standard deviation divided by the square root of the sample size (SE = \(\sigma / \sqrt{n}\)). Then we use the Z-score formula to find the Z-scores that correspond to the mean \(\pm\) 5.11 liters. The Z-scores are calculated as \(Z = (X-\mu) / SE\).

After we have the Z-scores, we look up the corresponding probabilities in the standard normal distribution table and subtract the smaller probability from the larger probability to find the probability of being within \(\pm\) 5.11 liters of the mean. Lastly, because the Central Limit Theorem applies, proper analysis using the standard normal distribution will yield an accurate result for this problem.

Answer 2

The probability that the sample mean differs from the true mean by less than 5.11 liters is approximately 92.66%, calculated using the Central Limit Theorem and standard normal distribution tables.

Step-by-step explanation:

To find the probability that the sample mean would differ from the true mean by less than 5.11 liters, we can use the Central Limit Theorem since the sample size is large. Given that the mean per capita consumption of milk per year is 153 liters with a standard deviation of 27 liters, and the sample size is 90, we can calculate the standard error (SE) of the sample mean which is given by SE = σ/√n, where σ is the standard deviation and n is the sample size.

Substituting the given values: SE = 27 liters/√90 ≈ 2.846 liters.

To find the z-scores corresponding to ±5.11 liters from the mean, we use the formula z = (X - μ)/SE, where μ is the mean and X is the value for which we are finding the z-score.

For the upper limit, z = (153 + 5.11 - 153)/2.846 ≈ 1.795, and for the lower limit, z = (153 - 5.11 - 153)/2.846 ≈ -1.795.

We now look up these z-scores in the standard normal distribution table, which gives us the probabilities for each z-score. The probability that the sample mean is between these z-scores is the difference in the probabilities corresponding to the z-scores, which is essentially the area under the normal curve between these z-scores.

Consulting the standard normal distribution table, the probabilities corresponding to ±1.795 are approximately 0.9633 and 0.0367, respectively. Hence, the probability that the sample mean would differ from the true mean by less than 5.11 liters is 0.9633 - 0.0367 = 0.9266 or 92.66%.