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 $80\mathrm{\%}$ confidence interval for the mean number of letter sounds identified in one minute. Round the answers to at least two decimal places.
An $80\mathrm{\%}$ 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 $95\mathrm{\%}$ 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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Step 1 ：Given that the sample mean ($\overline{x}$) 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 $\overline{x}\pm Z\frac{s}{\sqrt{n}}$.

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 $\boxed{{\displaystyle 31.48<\mu <36.62}}$.

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.