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The image you uploaded provides a statistics problem involving the weights of newborn girls and asks for specific statistical calculations based on the summary statistics provided:

- \( n = 36 \) (sample size)
- \( \bar{x} = 3180.6 \, g \) (sample mean)
- \( s = 700.5 \, g \) (sample standard deviation)
- A confidence level of 95%.

Here are the steps for parts a, b, and c based on the image:

### a. Identify the critical value \( t_{\alpha/2} \)
- The critical value is given in the image as:
  \[
  t_{\alpha/2} = 2.03
  \]
  This is based on a 95% confidence level and the degrees of freedom \( n - 1 = 35 \).

### b. Find the margin of error
The margin of error \( E \) is calculated using the formula:
\[
E = t_{\alpha/2} \times \left( \frac{s}{\sqrt{n}} \right)
\]
Substituting the values:
\[
E = 2.03 \times \left( \frac{700.5}{\sqrt{36}} \right)
\]
First, calculate the standard error:
\[
\frac{700.5}{\sqrt{36}} = \frac{700.5}{6} = 116.75
\]
Now, calculate the margin of error:
\[
E = 2.03 \times 116.75 = 236.0 \, g
\]
This matches the margin of error given in the image.

### c. Find the confidence interval estimate of \( \mu \)
The confidence interval is calculated as:
\[
\bar{x} - E < \mu < \bar{x} + E
\]
Substitute the values:
\[
3180.6 - 236.0 < \mu < 3180.6 + 236.0
\]
This simplifies to:
\[
2944.6 < \mu < 3416.6
\]

Thus, the confidence interval for the population mean \( \mu \) is \( 2944.6 \, g \) to \( 3416.6 \, g \).

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Would you like more details or have any questions? Here are some related questions:

1. How is the \( t \)-critical value chosen for different confidence levels?
2. How does sample size affect the margin of error?
3. Why do we use \( t \)-distribution instead of normal distribution for smaller sample sizes?
4. What is the relationship between standard deviation and margin of error?
5. How does increasing the confidence level affect the confidence interval?

**Tip**: The margin of error decreases as the sample size increases, as long as the standard deviation remains the same.

Here are summary statistics for randomly selected weights of newborn girls: n = 36, x̄ = 3180.6 g, s = 700.5 g. Use a confidence level of 95% to complete parts (a) through (d).

This problem calculates the confidence interval estimate of the mean weight of newborn girls using a sample of n = 36. Learn how to identify the critical value, calculate the margin of error, and construct the confidence interval. The problem uses the t-distribution with a 95% confidence level.

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