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. Listed below are brain volumes (cm $^{3}$ ) of twins from Data Set 8 "IQ and Brain Size" in Appendix B. Construct a $99 \%$ confidence interval estimate of the mean of the differences between brain volumes for the first-born and the second-born twins. What does the confidence interval suggest? $$\begin{array}{l|r|r|r|r|r|r|r|r|r|r} \hline \text { First Born } & 1005 & 1035 & 1281 & 1051 & 1034 & 1079 & 1104 & 1439 & 1029 & 1160 \\ \hline \text { Second Born } & 963 & 1027 & 1272 & 1079 & 1070 & 1173 & 1067 & 1347 & 1100 & 1204 \\ \hline \end{array}$$

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.
Listed below are brain volumes (cm $^{3}$ ) of twins from Data Set 8 "IQ and Brain Size" in Appendix B. Construct a $99 \%$ confidence interval estimate of the mean of the differences between brain volumes for the first-born and the second-born twins. What does the confidence interval suggest?
$$\begin{array}{l|r|r|r|r|r|r|r|r|r|r}
\hline \text { First Born } & 1005 & 1035 & 1281 & 1051 & 1034 & 1079 & 1104 & 1439 & 1029 & 1160 \\
\hline \text { Second Born } & 963 & 1027 & 1272 & 1079 & 1070 & 1173 & 1067 & 1347 & 1100 & 1204 \\
\hline
\end{array}$$

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Key Concepts

Paired Sample Data

Paired sample data refers to situations where two measurements are taken from the same subject or matched subjects. This design is used to control for individual variability by comparing the differences within each pair, making it easier to detect whether there is a systematic difference between the two conditions or treatments.

Confidence Interval

A confidence interval provides a range of values that is believed, with a specified degree of confidence (in this case 99%), to contain the true parameter of interest—in this situation, the mean difference between the paired measurements. It expresses the precision and uncertainty of the estimate based on the sample data.

t-Distribution

The t-distribution is used for inference when the sample size is small and the population standard deviation is unknown. It is particularly useful in paired comparisons, where the differences between paired observations are assumed to be normally distributed, allowing for valid construction of confidence intervals.

Normality Assumption

The assumption that the differences are approximately normally distributed is crucial for the reliability of the t-procedures applied in small samples. This assumption ensures that the confidence interval is accurate and that the inferential methods used remain valid.

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