A study was conducted at the VirginiaMaryland Regional...

A study was conducted at the VirginiaMaryland Regional College of Veterinary Medicine Equine Center to determine if the performance of a certain type of surgery on young horses had any effect on certain kinds of blood cell types in the animal. Fluid samples were taken from each of six foals before and after surgery. The samples were analyzed for the number of postoperative white blood cell (WBC) leukocytes. A preoperative measure of WBC leukocytes was also measured. The data are given as follows: $$ \begin{array}{ccc} \text { Foal } & \text { Presurgery* } & \text { Postsurgery* } \\ \hline 1 & 10.80 & 10.60 \\ 2 & 12.90 & 16.60 \\ 3 & 9.59 & 17.20 \\ 4 & 8.81 & 14.00 \\ 5 & 12.00 & 10.60 \\ 6 & 6.07 & 8.60 \\ \hline \end{array} $$ Use a paired sample $t$-test to determine if there is a significant change in WBC leukocytes with the surgery.

A study was conducted at the VirginiaMaryland Regional College of Veterinary Medicine Equine Center to determine if the performance of a certain type of surgery on young horses had any effect on certain kinds of blood cell types in the animal. Fluid samples were taken from each of six foals before and after surgery. The samples were analyzed for the number of postoperative white blood cell (WBC) leukocytes. A preoperative measure of WBC leukocytes was also measured. The data are given as follows:
$$
\begin{array}{ccc}
\text { Foal } & \text { Presurgery* } & \text { Postsurgery* } \\
\hline 1 & 10.80 & 10.60 \\
2 & 12.90 & 16.60 \\
3 & 9.59 & 17.20 \\
4 & 8.81 & 14.00 \\
5 & 12.00 & 10.60 \\
6 & 6.07 & 8.60 \\
\hline
\end{array}
$$
Use a paired sample $t$-test to determine if there is a significant change in WBC leukocytes with the surgery.

Key Concepts

Assumption of Normality

A key assumption underlying the paired t-test is that the differences between paired observations are normally distributed. This assumption ensures that the test's results are valid and that the t-distribution is an appropriate model for the test statistic, particularly when the sample size is small.

Hypothesis Testing Framework

The hypothesis testing framework in the paired t-test involves formulating a null hypothesis that there is no significant change between the paired measurements (i.e., the mean difference is zero) versus an alternative hypothesis that there is a significant change. This framework guides the calculation of the test statistic and the subsequent decision-making regarding the treatment effect.

Test Statistic and Degrees of Freedom

In the paired t-test, the test statistic is calculated by dividing the mean of the difference scores by the standard error of these differences. The distribution of the test statistic follows a t-distribution with degrees of freedom equal to the number of pairs minus one. These components are critical for determining whether the observed effect is statistically significant.

Paired Sample t-Test

The paired sample t-test is a statistical method used to compare two related means by analyzing the differences within paired observations. In this context, each subject (or foal) is measured before and after a treatment (surgery), allowing the test to control for the variability between subjects that might otherwise obscure the treatment effect. This analysis is crucial when observations are not independent but inherently linked.

Difference Score Calculation

Calculating the difference scores involves subtracting the pre-treatment measurement from the post-treatment measurement for each paired observation. These differences form the basis of analysis in a paired t-test and help quantify the individual changes attributable to the treatment, reducing the impact of between-subject variability.

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