Question

In a study of in vitro fertilization, 264 women ages \(40-44\) underwent a procedure known as elective single embryo transfer (eSET) to attempt to get pregnant. Sixty of these women successfully became pregnant and gave birth. \({ }^{14}\) Use these data to construct a \(95 \%\) confidence interval for the probability of success using eSET for a woman ages \(40-44\)

Short answer

The 95% confidence interval for the probability of success using eSET for a woman ages 40-44 is approximately 0.17 to 0.29.

Step-by-step solution

Identify the Sample Proportion

First, find the sample proportion (p-hat) which is the ratio of the number of successful pregnancies to the total number of women. The formula for the sample proportion is: \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes and \( n \) is the total sample size.

Compute the Standard Error

Calculate the standard error (SE) of the sample proportion using the formula: \( SE = \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n} } \). The standard error measures the variability of the sample proportion.

Determine the z-score for a 95% Confidence Interval

Find the z-score that corresponds to the middle 95% of the standard normal distribution. Typically, this value is approximately 1.96 for a 95% confidence interval.

Calculate the Confidence Interval

Use the formula for a confidence interval for a proportion: \( CI = \hat{p} \pm z \times SE \) to find the lower and upper bounds of the interval. Here, \( CI \) represents the confidence interval, \( \hat{p} \) is the sample proportion, \( z \) is the z-score found in step 3, and \( SE \) is the standard error calculated in step 2.

Interpret the Confidence Interval

The 95% confidence interval gives us a range of values within which we can be 95% confident that the true proportion of successes, for all women ages 40-44 using eSET, falls.

Key concepts

Sample Proportion

Understanding the concept of a sample proportion is crucial when delving into statistics, particularly concerning in vitro fertilization (IVF) success rates for women ages 40-44. The term sample proportion, often denoted as \( \hat{p} \), is a statistical measure that represents the fraction of the sample that has the characteristic of interest—in this case, the successful pregnancies following elective single embryo transfer (eSET).

For example, in the study mentioned, out of 264 women who underwent eSET, 60 became pregnant and gave birth. The sample proportion is therefore calculated as \( \hat{p} = \frac{60}{264} \). It is the estimator of the true population proportion, and understanding this figure helps researchers to draw conclusions about the broader population beyond the study participants.

Standard Error

The standard error (SE) is a measure that tells us how far the sample proportion may be from the true population proportion. In simple terms, it shows the precision with which a sample proportion estimates the true proportion. Higher standard error indicates less precision, while a lower standard error means higher accuracy.

To calculate the standard error for the sample proportion of successful pregnancies in the IVF study, one would use the formula: \( SE = \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n} } \), where \( \hat{p} \) is the sample proportion and \( n \) is the sample size. This calculation reflects the magnitude of uncertainty or 'margin of error' we can expect in estimating the true proportion from our sample data.

Z-Score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, using standard deviations. In the context of constructing confidence intervals, a z-score is used to capture the range of standard deviations that a certain percentage of data falls within.

In creating a 95% confidence interval, we look to the standard normal distribution (a bell curve), where roughly 95% of data lies within approximately 1.96 standard deviations on either side of the mean. This is the z-score for a 95% confidence level, providing the critical value which, when multiplied with the standard error, determines the width of the confidence interval on either side of the sample proportion.

In Vitro Fertilization Statistics

When examining in vitro fertilization statistics, confidence intervals are particularly informative. They give us a range within which we can expect the true success rates to fall, based on sample data. For women aged 40-44 undergoing eSET, the proportion of successful pregnancies is not just a single number, but rather an estimate within a range of values that suggest what the success rate might be for all women in this demographic group if we were to have information on the entire population.

This range is essential for medical professionals to give informed predictions and for potential patients to understand the likelihood of success. It also helps to set realistic expectations and policies around IVF treatment, which is increasingly important as couples often turn to IVF as a solution for fertility challenges.