Sound it out: Phonics is an instructional method in which children are taught to connect sounds with letters or groups of letters. A sample of 137 first-graders who were learning English were asked to identify as many letter sounds as possible in a period of one minute. The average number of letter sounds identified was 34.05 with a standard deviation of 23.51 .
Part 1 of 2
(a) Construct an $80 \%$ confidence interval for the mean number of letter sounds identified in one minute. Round the answers to at least two decimal places.
An $80 \%$ confidence interval for the mean number of letter sounds identified in one minute is $31.48< \mu< 36.62$.
Part: $1 / 2$
Part 2 of 2
(b) If a $95 \%$ confidence interval were constructed with these data, would it be wider or narrower than the interval constructed in Part (a)?
The confidence interval would be
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Answer
If a 95% confidence interval were constructed with these data, it would be wider than the interval constructed in Part (a) because a higher confidence level requires a larger margin of error.
Step 1 ：Given that the sample mean (\(\bar{x}\)) is 34.05, the sample standard deviation (\(s\)) is 23.51, and the sample size (\(n\)) is 137.
Step 2 ：We need to find the Z-score for an 80% confidence level. The Z-score for an 80% confidence level is 1.28.
Step 3 ：We use the formula for the confidence interval for the mean, which is \(\bar{x} \pm Z \frac{s}{\sqrt{n}}\).
Step 4 ：Substitute the given values into the formula to find the confidence interval: \(34.05 \pm 1.28 \frac{23.51}{\sqrt{137}}\).
Step 5 ：Calculate the margin of error, which is \(1.28 \frac{23.51}{\sqrt{137}} = 2.57\).
Step 6 ：Subtract the margin of error from the sample mean to find the lower limit of the confidence interval: \(34.05 - 2.57 = 31.48\).
Step 7 ：Add the margin of error to the sample mean to find the upper limit of the confidence interval: \(34.05 + 2.57 = 36.62\).
Step 8 ：Final Answer: An 80% confidence interval for the mean number of letter sounds identified in one minute is \(\boxed{31.48<\mu<36.62}\).
Step 9 ：If a 95% confidence interval were constructed with these data, it would be wider than the interval constructed in Part (a) because a higher confidence level requires a larger margin of error.
