Question

Example 6.6 .3 reports measurements of pain for children who have had their tonsils removed. Another variable measured in that experiment was the number of doses of Tylenol taken by the children in the two groups. Those data are $$\begin{array}{|lcc|}\hline & {\text { Type of surgery }} \\\\{ 2 - 3 } & \text { Conventional } & \text { Coblation } \\\\\hline n & 49 & 52 \\\\\bar{y} & 3.0 & 2.3 \\\\\text { SD } & 2.4 & 2.0 \\\\\hline\end{array}$$ Compute the standard error of \(\left(\bar{Y}_{1}-\bar{Y}_{2}\right)\).

Short answer

The standard error of \(\bar{Y}_{1} - \bar{Y}_{2}\) is approximately 0.4410.

Step-by-step solution

Understand the problem

We need to calculate the standard error of the difference in means, denoted as \(SE(\bar{Y}_{1} - \bar{Y}_{2})\). For independent samples T-test, the standard error of the difference between two means can be computed using the formula: \[ SE(\bar{Y}_{1} - \bar{Y}_{2}) = \sqrt{\frac{SD_1^2}{n_1} + \frac{SD_2^2}{n_2}} \] where \(SD_1\) and \(SD_2\) are the standard deviations for each group, and \(n_1\) and \(n_2\) are the sample sizes.

Insert the values into the formula

Substitute the given values into the formula: \[ SE(\bar{Y}_{1} - \bar{Y}_{2}) = \sqrt{\frac{2.4^2}{49} + \frac{2.0^2}{52}} \]

Compute the standard error

Perform the calculations within the square root: \[ SE(\bar{Y}_{1} - \bar{Y}_{2}) = \sqrt{\frac{5.76}{49} + \frac{4.0}{52}} \] \[ SE(\bar{Y}_{1} - \bar{Y}_{2}) = \sqrt{0.11755 + 0.07692} \] \[ SE(\bar{Y}_{1} - \bar{Y}_{2}) = \sqrt{0.19447} \] \[ SE(\bar{Y}_{1} - \bar{Y}_{2}) = 0.4410 \] (Rounded to four decimal places)

Key concepts

Independent Samples T-Test

An independent samples T-test, also known as a two-sample t-test, is a statistical procedure used to determine if there is a significant difference between the means of two independent groups. In the context of the exercise, it helps to compare the average number of Tylenol doses taken by children undergoing conventional surgery versus those undergoing coblation surgery.

To perform this test, several assumptions must be met, including that the samples are independent of each other, the data is normally distributed within each group, and the variances between groups are equal. When these assumptions are not met, other forms of analysis might be more appropriate. The formula to calculate the standard error of the difference between the means is crucial for determining the t-value in a T-test, which then helps assess whether the observed difference is statistically significant.

Difference in Means

The difference in means is a measure of the disparity between the average outcomes of two different groups. In the provided exercise, we are comparing the mean number of Tylenol doses for children post-tonsillectomy between conventional surgery and coblation. Calculating the difference in means alone doesn't indicate whether the observed difference is due to chance or represents a true difference between the groups.

This is where the T-test comes into play, using the standard error of the difference in means to help determine the likelihood that the two groups differ significantly from each other. Understanding the difference in means is important for researchers and practitioners in making informed decisions based on the data collected from their studies or experiments.

Standard Deviation

Standard deviation (SD) is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. In the textbook exercise, standard deviation plays a pivotal role as it reflects the variability of Tylenol doses taken by the children in each group.

When calculating the standard error of the difference in means, the standard deviations of each group are squared and divided by the respective sample sizes. The sum of these values is then square-rooted to obtain the standard error. This process highlights how the variability of each group affects the precision of the estimate of the difference in means.

Sample Size

Sample size refers to the number of observations or measurements taken from a population for the purposes of statistical analysis. In the context of calculating the standard error for an independent samples T-test, as seen in the exercise, the sample sizes of the two groups directly affect the magnitude of the standard error.

Larger sample sizes generally provide more accurate estimates of population parameters and, consequently, smaller standard errors, leading to more precise estimates of the difference in means. As the sample size increases, the confidence in the observed difference between the group means also increases, assuming all other factors are consistent. It is important for students to understand the impact of sample size not only on standard error but also on the overall validity of statistical tests, like the T-test.