A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 300 babies were​ born, and 270 of them were girls. Use the sample data to construct a 99​% confidence interval estimate of the percentage of girls born. Based on the​ result, does the method appear to be​ effective?
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Part 1
  
enter your response hereless than pless than
  
enter your response here ​(Round to three decimal places as​ needed.)

We can construct a 99% confidence interval for the percentage of girls born using the sample proportion method.

Given:
- Number of girls \( x = 270 \)
- Total number of babies \( n = 300 \)
- Confidence level = 99%

The formula for the confidence interval for a population proportion is:

\[
\hat{p} = \frac{x}{n}
\]

\[
\text{Confidence Interval} = \hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \( \hat{p} \) is the sample proportion
- \( z_{\alpha/2} \) is the z-score corresponding to the confidence level
- \( n \) is the total number of babies
- \( x \) is the number of girls

Step 1: Calculate \( \hat{p} \):

\[
\hat{p} = \frac{270}{300} = 0.9
\]

Step 2: Find the z-score for a 99% confidence interval. The value of \( z_{\alpha/2} \) for 99% confidence is approximately 2.576.

Step 3: Calculate the margin of error \( E \):

\[
E = 2.576 \times \sqrt{\frac{0.9 \times (1 - 0.9)}{300}} = 2.576 \times \sqrt{\frac{0.9 \times 0.1}{300}} = 2.576 \times \sqrt{0.0003}
\]

Step 4: Compute the confidence interval.

Let me calculate this.The 99% confidence interval for the percentage of girls born is approximately \( (0.855, 0.945) \), or 85.5% to 94.5%.

### Conclusion:
Since this confidence interval is well above the expected 50% (which would be the case if there was no effect of the method), the method appears to be effective in increasing the probability of conceiving a girl.

Would you like more details on any of the steps, or do you have any further questions?

### Related Questions:
1. What is the interpretation of a 99% confidence interval?
2. How does increasing the sample size affect the width of the confidence interval?
3. How would the interval change if the confidence level were 95% instead of 99%?
4. What assumptions are necessary for the confidence interval calculation to be valid?
5. How does the z-score affect the confidence interval?

### Tip:
A higher confidence level gives a wider confidence interval, reflecting greater uncertainty in the estimate.

A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study, 300 babies were born, and 270 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective?

Construct a 99% confidence interval estimate for the percentage of girls born in a clinical trial where 270 out of 300 babies were girls. This solution explores the sample proportion method and confidence intervals for population proportions. Ideal for high school students studying statistics.

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