The data in the next column represent the age (in weeks) at...

The data in the next column represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby. In Problem 34 from Section $9.2,$ it was verified that the data are normally distributed and that $s=10.00$ weeks. Construct and interpret a $95 \%$ confidence interval for the population standard deviation of the age (in weeks) at which babies first crawl. $$ \begin{array}{llllll} \hline 52 & 30 & 44 & 35 & 39 & 26 \\ \hline 47 & 37 & 56 & 26 & 39 & 28 \\ \hline \end{array} $$

The data in the next column represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby. In Problem 34 from Section $9.2,$ it was verified that the data are normally distributed and that $s=10.00$ weeks. Construct and interpret a $95 \%$ confidence interval for the population standard deviation of the age (in weeks) at which babies first crawl.
$$
\begin{array}{llllll}
\hline 52 & 30 & 44 & 35 & 39 & 26 \\
\hline 47 & 37 & 56 & 26 & 39 & 28 \\
\hline
\end{array}
$$

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Key Concepts

Confidence Interval for a Population Standard Deviation

A confidence interval for a population standard deviation is an interval estimate that provides a range of likely values for the true standard deviation. This is typically constructed based on the chi-square distribution when the sample is drawn from a normally distributed population, and it takes into account the variability of the sample variance to infer about the population parameter.

Chi-Square Distribution

The chi-square distribution is used to model the distribution of the sum of squared standard normal variables and is essential in constructing confidence intervals for the variance or standard deviation. In the context of normally distributed data, the sample variance multiplied by a factor based on the sample size follows a chi-square distribution, allowing for the estimation of confidence intervals for the population variance.

Degrees of Freedom

Degrees of freedom refer to the number of values in a calculation that are free to vary when estimating statistical parameters. For constructing a confidence interval for the variance or standard deviation, the degrees of freedom are typically the sample size minus one. This number is critical in determining the appropriate chi-square critical values used in the interval estimation process.

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