Sure, let's go through the detailed steps to solve this problem using the sample data provided.
### Step 1: Determine the Sample Proportion
First, we need to find the sample proportion ([tex]\(\hat{p}\)[/tex]) of girls born in the study. The proportion can be calculated as:
[tex]\[ \hat{p} = \frac{\text{number of girls}}{\text{total number of babies}} \][/tex]
Given that 286 out of 520 babies were girls:
[tex]\[ \hat{p} = \frac{286}{520} = 0.55 \][/tex]
### Step 2: Calculate the Standard Error
The formula for the standard error (SE) of the sample proportion is:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
where [tex]\( \hat{p} = 0.55 \)[/tex] and [tex]\( n = 520 \)[/tex].
The calculated standard error is:
[tex]\[ SE = \sqrt{\frac{0.55 \times (1 - 0.55)}{520}} \approx 0.021817 \][/tex]
### Step 3: Determine the Z-Score for 99% Confidence Level
To construct a 99% confidence interval, we look up the z-score corresponding to a 99% confidence level, which is approximately:
[tex]\[ z = 2.576 \][/tex]
### Step 4: Calculate the Margin of Error
The margin of error (ME) is calculated using the formula:
[tex]\[ ME = z \times SE \][/tex]
Using the values obtained:
[tex]\[ ME = 2.576 \times 0.021817 \approx 0.056196 \][/tex]
### Step 5: Construct the Confidence Interval
Finally, we can construct the 99% confidence interval for the proportion of girls born. The formula for the confidence interval (CI) is:
[tex]\[ \text{CI} = (\hat{p} - ME, \hat{p} + ME) \][/tex]
Substitute the values:
[tex]\[ \text{CI} = (0.55 - 0.056196, 0.55 + 0.056196) \][/tex]
This simplifies to:
[tex]\[ \text{CI} = (0.494, 0.606) \][/tex]
### Conclusion
The 99% confidence interval estimate for the percentage of girls born is [tex]\( (49.4\%, 60.6\%) \)[/tex].
### Interpretation
To conclude whether the method appears to be effective, we would typically compare this confidence interval to the natural birth rate of girls, which is approximately 50%. Since the lower bound of our interval (49.4%) is very close to but below 50% and the upper bound (60.6%) is above 50%, this interval suggests some evidence that the method could be effective in increasing the probability of having a girl. However, because part of the interval still falls below 50%, the evidence is not overwhelmingly strong. It would be more convincing if the entire interval were above 50%.
Thus, based on this interval alone, we can say there is some indication that the method may be effective, but further research would be needed to draw a more definitive conclusion.
