Observation 1 2 3 4 5 6 7 8 Xi 19.4 18.3 22.1 20.7

step 1 So we have our difference of i equaling the x of i minus the y sub i. And in part a, we had to l

Observation 1 2 3 4 5 6 7 8 Xi ...

$ \begin{array}{ccccccccc} \text { Observation } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline X_{i} & 19.4 & 18.3 & 22.1 & 20.7 & 19.2 & 11.8 & 20.1 & 18.6 \\ \hline Y_{i} & 19.8 & 16.8 & 21.1 & 22.0 & 21.5 & 18.7 & 15.0 & 23.9 \end{array} $ (a) Determine $d_{i}=X_{i}-Y_{i}$ for each pair of data. (b) Compute $\bar{d}$ and $s_{d}$ (c) Test if $\mu_{d} \neq 0$ at the $\alpha=0.01$ level of significance. (d) Compute a $99 \%$ confidence interval about the population mean difference $\mu_{d}$.

$
\begin{array}{ccccccccc}
\text { Observation } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\
\hline X_{i} & 19.4 & 18.3 & 22.1 & 20.7 & 19.2 & 11.8 & 20.1 & 18.6 \\
\hline Y_{i} & 19.8 & 16.8 & 21.1 & 22.0 & 21.5 & 18.7 & 15.0 & 23.9
\end{array}
$
(a) Determine $d_{i}=X_{i}-Y_{i}$ for each pair of data.
(b) Compute $\bar{d}$ and $s_{d}$
(c) Test if $\mu_{d} \neq 0$ at the $\alpha=0.01$ level of significance.
(d) Compute a $99 \%$ confidence interval about the population mean difference $\mu_{d}$.

Key Concepts

Paired Data Analysis

Paired data involves collecting two related measurements from each subject or experimental unit. This design is used when comparing two treatments, time points, or conditions, allowing each pair to serve as its own control. The primary goal is to reduce the variability that can occur when using independent groups, thereby focusing on the inherent difference within each pair.

Difference Calculation in Paired Data

In paired data analysis, the first step is often to compute the difference between the two measurements for each pair. This transformation simplifies the analysis by reducing the two sets of correlated data into a single set of differences, which directly reflects the effect or change being studied.

Sample Mean and Standard Deviation of Differences

Once the differences are computed, the sample mean of these differences estimates the average change or effect, while the sample standard deviation of the differences quantifies the variability among the pairs. These two statistics are crucial for further inferential procedures, such as constructing confidence intervals or performing hypothesis tests.

Paired t-test

The paired t-test is a statistical method used to determine whether the mean difference between paired observations is statistically significantly different from a specific value, typically zero. This test takes into account the paired nature of the data by considering the differences, and it uses the t-distribution with degrees of freedom equal to the number of pairs minus one.

Hypothesis Testing

Hypothesis testing involves the formulation of a null hypothesis, typically stating that there is no effect or difference, and an alternative hypothesis indicating a significant effect. In the context of a paired t-test, the null hypothesis posits that the population mean difference is zero. The test statistic, computed using the sample mean difference and its standard deviation, is then compared to critical values from the t-distribution at a chosen significance level.

Confidence Interval for the Mean Difference

A confidence interval provides a range of values within which the true population mean difference is likely to lie, with a certain level of confidence (e.g., 99%). It is calculated using the sample mean difference, its standard deviation, and appropriate t-distribution critical values. This interval offers insight into the precision of the estimated effect and guides conclusions about the statistical significance of the observed differences.

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