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Lesson: 8.1 Estimating Population Means (...
LACARA OWENS
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A marketing research company desires to know the mean consumption of milk per week among males over age 25 . They believe that the milk consumption has a mean of 2.5 liters, and want to construct a $85 \%$ confidence interval with a maximum error of 0.07 liters. Assuming a variance of 1.21 liters, what is the minimum number of males over age 25 they must include in their sample? Round your answer up to the next integer.

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Step 1 ：A marketing research company wants to know the average milk consumption per week among males over the age of 25. They believe that the mean milk consumption is 2.5 liters, and they want to construct an 85% confidence interval with a maximum error of 0.07 liters. Assuming a variance of 1.21 liters, we need to find the minimum number of males over the age of 25 that they must include in their sample. We will round our answer up to the next integer.

Step 2 ：We start by identifying the given values. The z-score for an 85% confidence level is \(Z = 1.44\). The standard deviation, which is the square root of the variance, is \(\sigma = \sqrt{1.21} = 1.1\). The maximum error is \(E = 0.07\).

Step 3 ：We can calculate the sample size using the formula \(n = \frac{Z^2 \cdot \sigma^2}{E^2}\).

Step 4 ：Substituting the given values into the formula, we get \(n = \frac{(1.44)^2 \cdot (1.1)^2}{(0.07)^2}\).

Step 5 ：Calculating the above expression, we get \(n = 513\). However, since we cannot have a fraction of a person, we round up to the next integer, which gives us \(n = 513\).

Step 6 ：Final Answer: The minimum number of males over the age of 25 they must include in their sample is \(\boxed{513}\).