Use the listed paired sample data, and assume that the...

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have 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 (in.) and measured heights (in.) for males aged $12-16 .$ 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 $0 .$ $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Reported } & 68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72 \\ \hline \text { Measured } & 67.9 & 69.9 & 64.9 & 68.3 & 70.3 & 60.6 & 64.5 & 67.0 & 55.6 & 74.2 & 65.0 & 70.8 \\ \hline \end{array}$$

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have 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 (in.) and measured heights (in.) for males aged $12-16 .$ 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 $0 .$
$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \text { Reported } & 68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72 \\
\hline \text { Measured } & 67.9 & 69.9 & 64.9 & 68.3 & 70.3 & 60.6 & 64.5 & 67.0 & 55.6 & 74.2 & 65.0 & 70.8 \\
\hline
\end{array}$$

Key Concepts

Paired Sample Data

In statistics, paired sample data refers to a set of observations where each data point in one sample is uniquely matched with a data point in another sample. This design is commonly used to compare two treatments or measurements taken from the same subjects under different conditions, thereby controlling for individual variability. It is crucial when the focus is on the differences within each pair rather than independent samples.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true parameter of interest—in this case, the mean difference. The 99% confidence interval indicates that, if the same population were sampled numerous times, approximately 99% of the intervals calculated from those samples would include the actual mean difference, providing a measure of the estimate's reliability.

t-Distribution

When dealing with small sample sizes or when the population standard deviation is unknown, the t-distribution is used to construct confidence intervals or perform hypothesis tests. The t-distribution adjusts for the additional variability expected in estimates from small samples, and its shape depends on the degrees of freedom, which in paired data is often the number of pairs minus one.

Statistical Significance and Interpretation of Zero Within the Confidence Interval

The incorporation (or exclusion) of zero in the confidence interval for the mean difference has important implications. If zero is not included, it suggests that there is a statistically significant difference between the paired measurements at the chosen confidence level. Conversely, if zero is within the interval, there isn’t enough evidence to conclude a significant difference, meaning any observed difference may be due to random variation.

Normality Assumption

The assumption of approximate normality in the distribution of differences is essential when employing parametric methods like the paired t-test. This assumption allows for the application of the t-distribution to estimate the confidence interval. It is particularly important in small sample sizes because deviations from normality can lead to misleading inferences.

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