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Chapter 6: Problem 126

Using the dataset NutritionStudy, we calculate that the average number of grams of fat consumed in a day for the sample of \(n=315\) US adults in the study is \(\bar{x}=77.03\) grams with \(s=33.83\) grams. (a) Find and interpret a \(95 \%\) confidence interval for the average number of fat grams consumed per day by US adults. (b) What is the margin of error? (c) If we want a margin of error of only 卤1 , what sample size is needed?

Short Answer

Expert verified
The 95% confidence interval for the average number of fat grams consumed per day by US adults is \(77.03 \pm 1.96\times \dfrac{33.83}{\sqrt{315}}\) grams. The margin of error for this data is \(1.96\times\dfrac{33.83}{\sqrt{315}}\) grams. To achieve a margin of error of 卤1 gram, a sample size of \((\dfrac{1.96 \times 33.83}{1})^2\) or approximately 1124 people is needed.

Step by step solution

01

Calculate the 95% Confidence interval

A 95% confidence interval can be calculated using the T distribution since the population standard deviation is unknown. We'll use the formula \(\bar{x} \pm t_{\alpha/2, n-1} \cdot \dfrac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(t_{\alpha/2, n-1}\) is the t score for a desired level of confidence with \(n-1\) degrees of freedom, \(s\) is the sample standard deviation, and \(n\) is the sample size. The t score for 95% confidence (2-tailed) and \(n-1 = 315-1 = 314\) degrees of freedom is approximately 1.96. Substituting these values we have \(77.03 \pm 1.96\times \dfrac{33.83}{\sqrt{315}} \) grams.
02

Compute the Margin of Error

The margin of error is the part of the confidence interval calculation that adds and subtracts from the mean. It is calculated with the formula \(t_{\alpha/2, n-1} \cdot \dfrac{s}{\sqrt{n}}\) . With the given sample data, the margin of error calculation would be \(1.96\times\dfrac{33.83}{\sqrt{315}}\) grams.
03

Find the Required Sample Size

For a desired margin of error (\(E\)) of only 卤1 gram, we can rearrange the margin of error formula to solve for \(n\), the sample size. The formula becomes \(n = (\dfrac{t_{\alpha/2, n-1} \cdot s}{E})^2\). Using the given values, the sample size would be \((\dfrac{1.96 \times 33.83}{1})^2\). We always round up to the next whole number when determining a sample size because we can't have a fraction of a person.

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