A clinical trial tests a method designed to increase the probability of conceiving a
girl. In the study 395 babies were born, and 316 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?
To construct a confidence interval estimate of the percentage of girls born, we can use the formula for the confidence interval for a proportion.
The formula for the Confidence Interval for a Proportion is given by:
CI = p̂ ± z * sqrt((p̂ * (1 - p̂)) / n),
where p̂ is the sample proportion, z is the critical value (corresponding to the desired level of confidence), sqrt is the square root, and n is the sample size.
In this case, we have n = 395 babies and a total of 316 girls born.
First, we calculate the sample proportion, p̂, by dividing the number of girls by the total number of babies:
p̂ = 316 / 395 ≈ 0.8
Next, we need to find the critical value, z, for a 99% confidence interval. Since the confidence level is 99%, the alpha level (α) is given by (1 - confidence level), which is 1 - 0.99 = 0.01. We divide this by 2 to get a two-tailed test and find the z-value associated with an alpha of 0.005 in the standard normal distribution. The z-value can be found using a z-table or a calculator, and for a 99% confidence level, the z-value is approximately 2.576.
Now, we can plug in the values into the formula to calculate the confidence interval:
CI = 0.8 ± 2.576 * sqrt((0.8 * (1 - 0.8)) / 395)
Calculating the square root and performing the remaining calculations, we get:
CI ≈ 0.8 ± 0.032
Therefore, the 99% confidence interval estimate of the percentage of girls born is approximately 0.768 to 0.832.
Now, to determine whether the method appears to be effective, we can check if the confidence interval includes the value of 0.5 (50%). If the interval includes 0.5, it suggests that the method did not have a significant effect. However, if the interval does not include 0.5, it indicates a significant effect.
Since the confidence interval (0.768 to 0.832) does not include 0.5, it suggests that the method is effective in increasing the probability of conceiving a girl.
Please note that this conclusion is based on the given data and assumptions of the study. Further analysis and considerations may be required for a more definitive answer.