Large Data Sets from Appendix B. In Exercises 21 and 22, use the data set in Appendix B. Assume that each sample is a simple random sample obtained from a population with a normal distribution.


Birth Weights Refer to Data Set 6 “Births” in Appendix B.


a. Use the 205 birth weights of girls to construct a 95% confidence interval estimate of the standard deviation of the population from which the sample was obtained.

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with constructing a 95% confidence interval for the standard deviation of the population using the sample of 205 birth weights of girls. This involves using the chi-square distribution, as confidence intervals for population variance or standard deviation are based on this distribution.
2. Step 2: Identify the formula for the confidence interval of the population variance. The formula is: \( \left( \frac{(n-1)s^2}{\chi^2_{\text{upper}}}, \frac{(n-1)s^2}{\chi^2_{\text{lower}}} \right) \), where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) are the critical values of the chi-square distribution corresponding to the desired confidence level.
3. Step 3: Calculate the sample variance \( s^2 \) using the birth weights data. The formula for variance is \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \), where \( x_i \) are the individual data points, \( \bar{x} \) is the sample mean, and \( n \) is the sample size. Compute \( s^2 \) using the provided data set.
4. Step 4: Determine the critical values \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) for the chi-square distribution. Use the degrees of freedom \( df = n-1 \) and the confidence level (95%) to find these values from a chi-square table or statistical software. The critical values correspond to the upper and lower tails of the distribution.
5. Step 5: Plug the values into the formula for the confidence interval of the variance. Once the confidence interval for the variance is calculated, take the square root of both bounds to obtain the confidence interval for the standard deviation. The final interval will provide the range within which the population standard deviation is likely to fall with 95% confidence.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a data set, 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 we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population parameter.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of a population, it quantifies how much individual data points deviate from the mean. A smaller standard deviation indicates that the data points tend to be closer to the mean, while a larger standard deviation indicates more spread out values.

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger population, where each individual has an equal chance of being selected. This method helps ensure that the sample is representative of the population, reducing bias and allowing for valid statistical inferences about the population based on the sample data.