Question

A group of female college students were divided into three groups according to upper body strength. Their leg strength was tested by measuring how many consecutive times they could leg press 246 pounds before exhaustion. (The subjects were allowed only one second of rest between consecutive lifts.) The data are shown in the following table. \({ }^{25}\) What type of analysis is appropriate for these data? $$ \begin{array}{|lccc|} \hline && {\text { Upper body strength group }} \\ & \text { Low } & \text { Middle } & \text { High } \\ \hline & 55 & 40 & 181 \\ & 70 & 200 & 85 \\ & 45 & 250 & 416 \\ & 246 & 192 & 228 \\ & 240 & 117 & 257 \\ & 96 & 215 & 316 \\ & 225 & & 134 \\ \hline \text { Mean } & 140 & 169 & 231 \\ \text { SD } & 93 & 77 & 112 \\ \hline \end{array} $$

Short answer

An Analysis of Variance (ANOVA) test is appropriate for these data to compare the means across the three independent groups based on upper body strength.

Step-by-step solution

Identifying the Type of Data

Examine the table provided to determine the nature of the data. The data consists of numerical values for the number of consecutive leg presses, which can be considered continuous. Also, the data is grouped into three categories based on upper body strength ('Low', 'Middle', 'High'), which suggests that these are independent groups.

Choosing the Appropriate Analysis

Since the data represents continuous numerical values and involves comparing means across three independent groups, an Analysis of Variance (ANOVA) test would be appropriate. The test can determine if there is a statistically significant difference in the mean number of leg presses among the three different upper body strength groups.

Key concepts

Continuous Data

When we talk about continuous data, we're referring to quantitative information that can take on any value within a given range. Imagine a smooth scale where you can mark any point, not just whole numbers—that's continuous data. In the context of our exercise, the number of times the students can leg press 246 pounds is a perfect example. This data is not restricted to certain fixed values; rather, it can include fractions and decimals, emphasizing the continuous nature. It's the kind of data that is measured, not counted, and is essential for conducting certain statistical analyses.
For example, the leg press counts could theoretically include half presses, or even more precise measurements, demonstrating the continuity. It is this characteristic that makes tools like Analysis of Variance (ANOVA) particularly useful because they can account for the nuances in such data.

Independent Groups

Independent groups refer to separate categories that do not influence each other—each group is distinct and does not overlap. In our scenario, female college students were split into 'Low', 'Middle', and 'High' upper body strength groups. These students have been categorized based on their upper body strength and we assume that each group has no bearing on the performance or results of the other groups. This is crucial for our ANOVA test because it relies on the assumption that groups are independent for accurate interpretation.
When conducting an ANOVA, it's important that the groups are indeed independent, because any form of relatedness or influence between the group members could skew the results and lead to incorrect conclusions. The independence of groups helps ensure that variation within the data is due to the variable being tested (in this case, upper body strength), rather than confounding factors.

Upper Body Strength

In our study, upper body strength is what we call the 'independent variable'—the factor we think might influence the outcome, which is the number of consecutive leg presses. How strong these students' upper bodies are could theoretically affect their leg strength and endurance, though it might not be immediately apparent why. This is a hypothesis we have that needs to be tested with the ANOVA.
It's interesting to consider why we're looking at upper body strength when measuring leg strength. There could be various reasons for this: perhaps we hypothesize that overall physical fitness relates to both upper and lower body, or that those with stronger upper bodies engage in more comprehensive workout routines that also improve leg strength. By categorizing the participants into different levels of upper body strength, we aim to see if there's a correlation between this and their leg press performance.

Statistical Significance

A key term in any analysis is 'statistical significance'. It's not enough for our ANOVA to simply find a difference in leg press counts between the groups; we need to know if this difference is not likely due to chance. Statistical significance tells us just that—if a result is statistically significant, it means that the findings are unlikely to have occurred randomly, and there's a specific probability (often less than 5%) that the observed effects are real.
The p-value we obtain from an ANOVA test indicates the likelihood of the data if there were no actual difference between our groups. If this p-value is below the pre-determined significance level (often set at 0.05), we reject the null hypothesis, concluding that there is indeed a statistically significant difference between upper body strength groups in terms of leg press performance.

Mean Comparison

At the heart of the ANOVA lies the comparison of means—or averages. By calculating and comparing the average number of leg presses for each level of upper body strength, ANOVA assesses whether the group means differ more than we would expect by chance. It's not looking at individual scores, but rather at the general pattern across the groups.
The beauty of ANOVA is that it can compare more than two groups at the same time, and in our case, we have three: 'Low', 'Middle', and 'High'. This is particularly helpful because it allows researchers to get a more complete picture of the relationship between upper body strength and leg press performance. If the ANOVA shows a significant result, it suggests that the mean number of leg presses is significantly different for at least one group compared to the others, hinting at a potential influence of upper body strength on this particular measure of leg endurance.