Question 21 An automobile manufacturer needs to estimate the average highway fuel economy (miles per gallon) for a new truck that will be sold starting next year. A random sample of 109 trucks were tested for highway fuel economy and the sample had a mean of 28.7 miles per gallon with a standard deviation of 2.8 miles per gallon. Using a $98 \%$ confidence level, determine the margin of error, $E$, and a confidence interval for the average highway fuel economy of this new truck. Report the confidence interval using interval notation. Round solutions to two decimal places, if necessary. The margin of error is given by $E=\square$. A $98 \%$ confidence interval is given by Question Help: $D$ Video Submit All Parts $75^{\circ} \mathrm{F}$ Mostly sunny Search
Solution
Step 1 ：The question is asking for the margin of error and the confidence interval for the average highway fuel economy of a new truck. The margin of error can be calculated using the formula \(E = Z * (\sigma/\sqrt{n})\), where \(Z\) is the Z-score corresponding to the desired confidence level, \(\sigma\) is the standard deviation, and \(n\) is the sample size. The confidence interval can then be calculated as \((\bar{x} - E, \bar{x} + E)\), where \(\bar{x}\) is the sample mean.
Step 2 ：Given: Sample size (n) = 109, Sample mean (\(\bar{x}\)) = 28.7 miles per gallon, Standard deviation (\(\sigma\)) = 2.8 miles per gallon, Confidence level = 98%
Step 3 ：We need to find the Z-score corresponding to a 98% confidence level. This can be found using a Z-table or a statistical calculator. The Z-score for a 98% confidence level is approximately 2.33.
Step 4 ：Let's calculate the margin of error and the confidence interval.
Step 5 ：n = 109, \(\bar{x}\) = 28.7, \(\sigma\) = 2.8, Z = 2.33
Step 6 ：Calculate the margin of error: \(E = Z * (\sigma/\sqrt{n}) = 2.33 * (2.8/\sqrt{109}) = 0.62\) (rounded to two decimal places)
Step 7 ：Calculate the confidence interval: \((\bar{x} - E, \bar{x} + E) = (28.7 - 0.62, 28.7 + 0.62) = (28.08, 29.32)\) (rounded to two decimal places)
Step 8 ：Final Answer: The margin of error is \(E = \boxed{0.62}\) (rounded to two decimal places). The 98% confidence interval is given by \((\boxed{28.08}, \boxed{29.32})\) (rounded to two decimal places).
