Nine subjects, 87 to 96 years old, were given 8 weeks of...

Nine subjects, 87 to 96 years old, were given 8 weeks of progressive resistance weight training. Strength before and after training for each individual was measured as maximum weight (in kilograms) lifted by left knee extension: $$ \begin{array}{llllllllllll} & & & & & & & & & & \text { Mean } & \text { \& } \\ \text { Before: } & 3 & 3.5 & 4 & 6 & 7 & 8 & 8.5 & 12.5 & 15 & 7.5 & 4 \\ \text { After: } & 7 & 17 & 19 & 12 & 19 & 22 & 28 & 20 & 28 & 19.1 & 6 . \\ \text { Difference: } & 4 & 13.5 & 15 & 6 & 12 & 14 & 19.5 & 7.5 & 13 & 11.6 & 4 \\ \hline \end{array} $$ Find a 95\% confidence interval estimate for the mean strength gain of all 87 - to 96 -year-olds who are given the training. (A) $11.6 \pm 3.03$ (B) $11.6 \pm 3.69$ (C) $11.6 \pm 3.76$ (D) $11.6 \pm 5.18$ (E) $11.6 \pm 5.70$

Nine subjects, 87 to 96 years old, were given 8 weeks of progressive resistance weight training. Strength before and after training for each individual was measured as maximum weight (in kilograms) lifted by left knee extension:
$$
\begin{array}{llllllllllll}
& & & & & & & & & & \text { Mean } & \text { \& } \\
\text { Before: } & 3 & 3.5 & 4 & 6 & 7 & 8 & 8.5 & 12.5 & 15 & 7.5 & 4 \\
\text { After: } & 7 & 17 & 19 & 12 & 19 & 22 & 28 & 20 & 28 & 19.1 & 6 . \\
\text { Difference: } & 4 & 13.5 & 15 & 6 & 12 & 14 & 19.5 & 7.5 & 13 & 11.6 & 4 \\
\hline
\end{array}
$$
Find a 95\% confidence interval estimate for the mean strength gain of all 87 - to 96 -year-olds who are given the training.
(A) $11.6 \pm 3.03$
(B) $11.6 \pm 3.69$
(C) $11.6 \pm 3.76$
(D) $11.6 \pm 5.18$
(E) $11.6 \pm 5.70$

Key Concepts

t-Distribution

The t-distribution is used instead of the normal distribution when the sample size is small and the population standard deviation is unknown. It accounts for additional uncertainty by having heavier tails, and is used in constructing confidence intervals and performing t-tests.

Difference Scores

In paired experiments, the analysis often involves computing the difference between paired measurements (such as after minus before values). These difference scores are then used to estimate the mean change and variability, which are crucial for constructing confidence intervals or conducting hypothesis tests.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of a paired t-test with n pairs, the degrees of freedom are n - 1, which is critical when determining the appropriate critical value from the t-distribution.

Confidence Interval

A confidence interval provides a range of plausible values for an unknown population parameter (such as the mean), based on sample data. It is constructed so that, in repeated samples, a certain percentage (e.g., 95%) of the intervals would contain the true parameter value.

Paired t-Test

The paired t-test is used when comparing two related samples, such as measurements taken before and after an intervention on the same subjects. It focuses on the differences within pairs to determine if there is a significant mean difference.

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