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

Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal. Self-Reported and Measured Male Heights 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. a. Is there sufficient evidence to support the claim that there is a difference between self reported heights and measured heights of males aged $12-16 ?$ Use a 0.05 significance level. b. Construct a $95 \%$ 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. Reported height $\quad\begin{array}{cccccccccccc}68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72\end{array}$ Measured height $\quad\begin{array}{cccccccccccc}67.9&69 .9&64 .9&68.3&70.3&60.6&64.5&67.0&55.6&74.2&65.0&70.8\end{array}$

Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal.
Self-Reported and Measured Male Heights 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.
a. Is there sufficient evidence to support the claim that there is a difference between self reported heights and measured heights of males aged $12-16 ?$ Use a 0.05 significance level.
b. Construct a $95 \%$ 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.
Reported height $\quad\begin{array}{cccccccccccc}68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72\end{array}$
Measured height $\quad\begin{array}{cccccccccccc}67.9&69 .9&64 .9&68.3&70.3&60.6&64.5&67.0&55.6&74.2&65.0&70.8\end{array}$

Key Concepts

Paired Sample t-test

This statistical test is used when comparing two means that come from the same group at different times or under different conditions. It focuses on the difference between paired observations in order to determine if there is a significant difference between the two measures, such as self-reported and measured heights, while accounting for the natural pairing in the data.

Null and Alternative Hypotheses

In the context of a paired t-test, the null hypothesis typically states that there is no difference between the paired measurements (i.e., the mean difference is zero), whereas the alternative hypothesis contends that there is a difference (i.e., the mean difference is not zero). These hypotheses are foundational for testing if any observed difference is statistically significant.

Significance Level

The significance level, often denoted by alpha (?), is the threshold for determining whether the observed data are sufficiently inconsistent with the null hypothesis. An alpha level of 0.05 means that there is a 5% risk of rejecting the null hypothesis when it is actually true, which sets the bar for deeming the differences statistically significant during the hypothesis test.

95% Confidence Interval

A 95% confidence interval provides a range of values for the mean difference between the paired measurements such that there is a 95% probability that the interval contains the true mean difference. In hypothesis testing for paired data, this interval offers an estimate of the magnitude and direction of the difference between the two measures.

Interpretation of Confidence Interval Results

Interpreting a confidence interval involves assessing whether the range includes zero. If zero is not within the interval, it suggests that the difference between the paired measurements is statistically significant at the chosen significance level. Conversely, if zero falls within the interval, it indicates insufficient evidence to conclude a true difference exists, and the observed difference might be due to chance.

Assumptions of the Test

The paired sample t-test is based on several assumptions: the paired differences should be approximately normally distributed, the data pairs should be randomly sampled, and the differences should be independent of each other. These assumptions are crucial for ensuring the validity of the test's results and conclusions.

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