Assume that the paired sample data are simple random samples...

Assume that the paired sample data are simple random samples and tbat the differences bave a distribution that is approximately normal. As part of the National Health and Nutrition Examination Survey, the Department of Health and Human Services obtained self-reported heights and measured heights for males aged $12-16 .$ All measurement are in inches. Listed below are sample results. Construct a $99 \%$ confidence interval estimate of the mean difference between reported heights and measured heights. Interpret the resulting confidence interval, and comment on the implications of whether the confidence interval limits contain $0 .$ $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline \text { Reported Height } & 68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72 \\ \hline \text { Measured Height } & 67.9 & 69.9 & 64.9 & 68.3 & 70.3 & 60.6 & 64.5 & 67.0 & 55.6 & 74.2 & 65.0 & 70.8 \\ \hline \end{array}$$

Assume that the paired sample data are simple random samples and tbat the differences bave a distribution that is approximately normal.
As part of the National Health and Nutrition Examination Survey, the Department of Health and Human Services obtained self-reported heights and measured heights for males aged $12-16 .$ All measurement are in inches. Listed below are sample results. Construct a $99 \%$ confidence interval estimate of the mean difference between reported heights and measured heights. Interpret the resulting confidence interval, and comment on the implications of whether the confidence interval limits contain $0 .$
$$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline \text { Reported Height } & 68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72 \\
\hline \text { Measured Height } & 67.9 & 69.9 & 64.9 & 68.3 & 70.3 & 60.6 & 64.5 & 67.0 & 55.6 & 74.2 & 65.0 & 70.8 \\
\hline
\end{array}$$

Key Concepts

Paired Sample Design

Paired sample design involves measuring two related observations on each subject and then analyzing the differences between these pairs. This method is used to reduce variability attributable to inherent differences among subjects, allowing for a more precise comparison between two conditions or measurement methods.

Confidence Interval Estimation

A confidence interval estimation provides a range within which the true population parameter is expected to lie with a certain probability (confidence level). In this context, a 99% confidence interval for the mean difference gives a range that is believed to contain the true average difference between the two paired measurements most of the time if similar sampling were repeated.

t-Distribution

The t-distribution is applied when the population standard deviation is unknown and the sample size is relatively small. It accounts for additional uncertainty in estimating the variability and is used for constructing confidence intervals and conducting hypothesis tests on the mean difference under the assumption that the differences are approximately normally distributed.

Interpretation of Confidence Interval Limits

Interpreting the confidence interval involves assessing whether the interval includes zero. If zero is not contained within the interval, it suggests that there is a statistically significant difference between the paired measurements, indicating a notable effect or discrepancy between the measured variables.

Assumptions of Normality and Simple Random Sampling

The analysis assumes that the paired differences are approximately normally distributed and that the data are collected via simple random sampling. These assumptions ensure the appropriateness of the t-distribution approximation and the overall reliability of the statistical inference drawn from the confidence interval.

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