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From the image, it appears that the problem involves constructing confidence intervals for the mean number of letter sounds identified by first-graders.

### Part (a): 99% Confidence Interval

You are provided with the following dataFrom the image, it appears that the problem involves constructing confidence intervals for the mean number of letter sounds identified by first-graders.

### Part (a): 99% Confidence Interval

You are provided with the following 
- Sample size \( n = 139 \)
- Sample mean \( \bar{x} = 34.12 \)
- Sample standard deviation \( s = 23.43 \)
- The interval calculated: \( 28.90 < \mu < 39.34 \)

The confidence interval was constructed using the formula for a confidence interval of the mean:
\[
\mu = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \( \bar{x} \) is the sample mean,
- \( s \) is the sample standard deviation,
- \( n \) is the sample size, and
- \( z \) is the critical value for a 99% confidence level.

### Part (b): Comparison of 99% and 90% Confidence Intervals

You are asked to determine whether a 90% confidence interval would be **wider** or **narrower** than the 99% interval.

For a 90% confidence interval, the critical value \( z \) would be smaller than for a 99% interval, as the 90% interval covers less area under the normal distribution curve.

Thus:
- A **90% confidence interval** would be **narrower** than the 99% interval because we are less confident and thus accept a smaller range of values for the population mean.

Would you like more detailed explanations for the calculations involved? 

Here are 5 related questions for further learning:
1. How is the critical value \( z \) determined for different confidence levels?
2. What is the relationship between confidence level and margin of error?
3. How does sample size affect the width of a confidence interval?
4. Why do we use the standard normal distribution for large samples in confidence interval calculations?
5. How would the confidence interval change if the sample standard deviation were smaller?

**Tip:** Increasing the confidence level will always result in a wider confidence interval, as you need to capture more possible values of the population parameter.

Construct a 99% confidence interval for the mean number of letter sounds identified in one minute by first-graders, and compare it to a 90% confidence interval.

Learn how to construct a 99% confidence interval for the mean number of letter sounds identified in one minute by first-graders in a phonics study. This problem explores confidence intervals, the z-distribution, and compares a 99% to a 90% confidence level. Suitable for students in advanced high school or introductory college statistics.

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