Question

Exercises 8.41 to 8.45 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } \\ \text { A } & 5 & 10.200 & 2.864 \\ \text { B } & 5 & 16.800 & 2.168 \\ \text { C } & 5 & 10.800 & 2.387\end{array}\) \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 2 & 133.20 & 66.60 & 10.74 & 0.002 \\ \text { Error } & 12 & 74.40 & 6.20 & & \\ \text { Total } & 14 & 207.60 & & & \end{array}\) What is the pooled standard deviation? What degrees of freedom are used in doing inferences for these means and differences in means?

Short answer

The pooled standard deviation is approximately 2.312439, and the degrees of freedom used are 14.

Step-by-step solution

Calculate the contribution of each group to the pooled standard deviation.

For group A: \((n_1 - 1) * s1^2 = (5-1) * 2.864^2 = 24.454656\). For group B: \((n_2 - 1) * s2^2 = (5-1) * 2.168^2 = 17.759744\). For group C: \((n_3 - 1) * s3^2 = (5-1) * 2.387^2 = 22.781769\).

Sum the group contributions

Total sum: \(24.454656 + 17.759744 + 22.781769 = 64.996169\).

Calculate the denominator of the pooled standard deviation formula

This is found by summing the sample sizes and subtracting the number of groups: \((n1 + n2 + n3 - k) = (5 + 5 + 5 - 3) = 12\).

Compute the pooled standard deviation

We do the square root of the contribution sum divided by the sum of the degrees of freedom: \(Sp = \sqrt{64.996169/12} = 2.312439\)

Find the Degrees of freedom

The degrees of freedom for inferences will equal to the total degrees of freedom shown in the table. Therefore, the degrees of freedom used for these means and differences in means is 14.

Key concepts

Degrees of Freedom

The concept of 'degrees of freedom' (DF) is essential in statistical analysis, acting as an integral part of various test and estimation procedures. In the context of pooled standard deviation, as in the textbook exercise, degrees of freedom represent the number of independent pieces of information available to estimate a statistic. When we calculate the variance or standard deviation for a single sample, we use DF = n - 1, where n is the sample size. This subtraction accounts for the fact that one value (the sample mean) has already been estimated, which restricts the variability of the dataset.

For ANOVA (Analysis of Variance), which compares the means of three or more groups, the calculation of degrees of freedom becomes partitioned. One part pertains to the variation between the groups (DF groups), and the other is concerned with the variation within the groups (DF error or DF within).

• DF groups is calculated as the number of groups minus one (k - 1).
• DF error is the total number of observations across all groups minus the number of groups (N - k).

The sum of these provides the total degrees of freedom, which is used to determine the critical value or p-value from the relevant distribution, be it a t-distribution or F-distribution, for the hypothesis being tested.

ANOVA

When analyzing datasets involving multiple groups, as seen in the textbook exercise, 'ANOVA' or Analysis of Variance is a statistical method used to test if there are significant differences between the means of three or more independent groups. ANOVA does this by comparing the ratio of between-group variance to within-group variance. The F-statistic, which results from this ratio, allows us to see if any of the group averages differ more than we'd expect by chance alone.

In the exercise, we are given an ANOVA table which includes several important metrics:

DF (Degrees of Freedom)

Which splits into 'Groups' and 'Error', helping us understand the number of independent values that can vary in the calculation of the statistics.

SS (Sum of Squares)

Which measures the total variation and splits again into 'Groups' (variation due to the interaction between groups) and 'Error' (variation within groups).

MS (Mean Square)

Which is calculated by dividing the 'SS' by their respective 'DF'. This average square gets larger if the observations are more spread out.

F

Which is the calculated test statistic from dividing 'MS Groups' by 'MS Error', and it tells us how much the group means vary relative to the variation within the groups themselves.
The 'P' value is then derived from the F-statistic, informing us whether the observed between-group differences are statistically significant. If the P-value is below a predetermined threshold (typically 0.05), we reject the null hypothesis that all group means are equal, suggesting there are significant differences among them.

Statistical Inference

Statistical inference involves drawing conclusions about a population's properties based on a sample. It encompasses a variety of procedures, including hypothesis testing, determining confidence intervals, and making predictions. In the case of our textbook exercise, statistical inference is used to decide whether the observed differences in sample means reflect true differences in population means or are merely a result of random sampling variation.

In the step-by-step solution, 'degrees of freedom' are particularly important for statistical inference as they influence the shape of the distribution used to determine the significance of our test statistic. For instance, a t-distribution, which is often used when estimating the mean of a normally-distributed population, becomes more similar to the standard normal distribution as the degrees of freedom increase.

• In our pooled standard deviation calculation, we use the degrees of freedom associated with the error term from the ANOVA table to pool the variance estimates.
• In hypothesis testing (such as in our ANOVA), degrees of freedom are also used to find the critical value or p-value associated with the test statistic.

Understanding statistical inference is crucial because it helps us make data-driven decisions and determine the reliability of our estimates and tests. Essentially, it's a measure of how confident we can be in the results obtained from our sample when applying them to the broader population.