Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Gender Selection Before its clinical trials were discontinued, the Genetics & IVF Institute conducted a clinical trial of the XSORT method designed to increase the probability of conceiving a girl and, among the 945 babies born to parents using the XSORT method, there were 879 girls. The YSORT method was designed to increase the probability of conceiving a boy and, among the 291 babies born to parents using the YSORT method, there were 239 boys. Construct the two 95% confidence interval estimates of the percentages of success. Compare the results. What do you conclude?

Verified step by step guidance

1. Step 1: Identify the problem and the data provided. For the XSORT method, we are given 945 total births and 879 girls. For the YSORT method, we are given 291 total births and 239 boys. The goal is to construct two 95% confidence intervals for the proportions of success (girls for XSORT and boys for YSORT).
2. Step 2: Calculate the sample proportions (p̂) for each method. For XSORT, the sample proportion is p̂ = 879 / 945. For YSORT, the sample proportion is p̂ = 239 / 291. These proportions represent the observed success rates for each method.
3. Step 3: Use the formula for the confidence interval for a population proportion: CI = p̂ ± Z * sqrt((p̂ * (1 - p̂)) / n), where Z is the critical value for a 95% confidence level (Z ≈ 1.96), p̂ is the sample proportion, and n is the sample size. Apply this formula separately for XSORT and YSORT.
4. Step 4: Plug in the values for XSORT. Use p̂ = 879 / 945 and n = 945. Calculate the margin of error (ME) using ME = Z * sqrt((p̂ * (1 - p̂)) / n). Then, compute the confidence interval as [p̂ - ME, p̂ + ME].
5. Step 5: Repeat the process for YSORT. Use p̂ = 239 / 291 and n = 291. Calculate the margin of error (ME) and the confidence interval as [p̂ - ME, p̂ + ME]. Finally, compare the two confidence intervals to determine if there is a significant difference in the success rates of the two methods.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically expressed as a percentage. For example, a 95% confidence interval suggests that if the same sampling method were repeated multiple times, approximately 95% of the calculated intervals would contain the true parameter. This concept is crucial for estimating proportions, such as the success rates of the XSORT and YSORT methods in this question.

Proportion

A proportion is a statistical measure that represents the part of a whole, often expressed as a fraction or percentage. In the context of this question, the proportions of girls born using the XSORT method and boys born using the YSORT method are essential for calculating the confidence intervals. Understanding how to calculate and interpret these proportions is vital for comparing the effectiveness of the two methods.

Statistical Comparison

Statistical comparison involves analyzing two or more groups to determine if there are significant differences between them. In this case, comparing the confidence intervals of the XSORT and YSORT methods allows us to assess whether the methods significantly differ in their success rates for gender selection. This concept is important for drawing conclusions based on the data and understanding the implications of the results.