Question

An experiment was conducted to investigate the effect of a training program on the length of time for a typical male college student to complete the 100 -yard dash. Nine students were placed in the program. The reduction \(y\) in time to complete the 100 -yard dash was measured for three students at the end of 2 weeks, for three at the end of 4 weeks, and for three at the end of 6 weeks of training. The data are given in the table. $$ \begin{array}{l|l|l|l} \text { Reduction in Time, } y(\mathrm{sec}) & 1.6, .8,1.0 & 2.1,1.6,2.5 & 3.8,2.7,3.1 \\ \hline \text { Length of Training, } x(\mathrm{wk}) & 2 & 4 & 6 \end{array} $$ Use an appropriate computer software package to analyze these data. State any conclusions you can draw.

Short answer

Answer: Based on the regression analysis, we can conclude that there is a positive relationship between the length of the training program and the reduction in time to complete the 100-yard dash, indicating that the training program is effective in improving students' performance. The strength of this relationship and its statistical significance can be further assessed through R-squared and p-values, respectively.

Step-by-step solution

Enter Data into Excel

Input the given data into an Excel sheet, where column A contains the Length of Training (x) data and column B contains the Reduction in Time (y) data.

Create a Scatterplot

Create a scatterplot to visualize the relationship between Length of Training (x) and Reduction in Time (y). This can be done in Excel by selecting the data, going to "Insert" tab, and selecting "Scatter" from Chart section.

Perform Regression Analysis

In Excel, go to the "Data" tab, click "Data Analysis," select "Regression," and click "OK." Fill in the input range for the x and y values. Make sure to check the "Label" checkbox if the column headers are included in the selection. Select an output range and click "OK."

Analyze Regression Output

Review the regression output provided by Excel. Pay attention to the values of the regression coefficients, R-squared, and the p-values. These will help to determine the strength of the relationship between the two variables and the statistical significance of the results.

Conclusions

Based on the regression analysis, we can draw the following conclusions: 1. There is a positive relationship between the length of the training program and the reduction in time to complete the 100-yard dash. This indicates that the training program is effective in improving students' performance. 2. The R-squared value gives us an indication of how well our model fits the data. The closer the R-squared value is to 1, the better our model fits the data. 3. The p-values can be used to determine if there is a statistically significant relationship between the length of the training program and the reduction in time to complete the 100-yard dash. Generally, a p-value less than 0.05 indicates a statistically significant relationship. The specific values for the R-squared, regression coefficients, and p-values will depend on the data and the software used for the analysis. Based on these findings, a conclusion can be made on the effectiveness of the training program on the student's performance in the 100-yard dash.

Key concepts

Understanding Scatterplots in Regression Analysis

A scatterplot is a type of graph used in statistics to display values for two variables. It allows you to visually assess the relationship between these variables. In the context of regression analysis, a scatterplot can help identify any potential correlations or patterns between the variables. For example, with the exercise data on the training program's effect on the 100-yard dash times, a scatterplot allows us to quickly see if there's a visible trend between more weeks of training and reduced dash times.

• Scatterplots consist of points plotted on a two-dimensional grid, where each point represents an observation from the dataset.
• The position of each dot is determined by the values of the two variables being compared.
• A line of "best fit" can be added to summarize the data trend in the scatterplot.

Creating a scatterplot in Excel involves selecting your data and choosing the scatterplot option. This visual tool is crucial for initial exploration of relationships in your data.

R-squared: Measuring the Fit of Regression Models

The R-squared value, also known as the coefficient of determination, evaluates how well your regression model fits the data. It gives an insight into the proportion of variance for the dependent variable that's explained by the independent variable in the model. An R-squared value close to 1 suggests that almost all of the variability in the outcome variable is captured by the independent variable.

• An R-squared value of 0 indicates that the model does not explain any of the variability.
• As the R-squared value increases towards 1, it indicates a better fit.
• Beyond measuring fit, a high R-squared does not imply causation and should be interpreted in context.

For the training program analysis, a suitable R-squared value would show that the duration of training has a strong relationship with the reduction in dash time. This would justify further steps, such as detailed statistical hypothesis testing.

Interpreting P-values in Statistical Analysis

P-values are a key concept in hypothesis testing in statistics. They help you determine whether the observed data falls under the assumed model or not. With a regression analysis, the p-value can tell you if the relationship between your independent variable and dependent variable is statistically significant.

• Typically, a p-value is compared against a significance level (often set at 0.05).
• A p-value lower than 0.05 usually suggests that the observed relationship is significant.
• Conversely, a p-value above 0.05 suggests a lack of statistically significant evidence to support the relationship.

In our exercise, if the p-values from the regression are below a threshold like 0.05, it indicates a statistically significant relationship between training duration and dash time reduction, thus confirming the effectiveness of the training program.

Utilizing Excel Data Analysis Tools for Regression

Excel's data analysis toolset is a powerful feature for performing tasks such as regression analysis efficiently. It offers easy-to-use wizards that can compute complex statistical operations like regression without requiring deep statistical expertise.

• Start by entering your data into an Excel spreadsheet, organizing them into columns.
• Use the "Data Analysis" toolpack to access regression capabilities by clicking on the "Data" tab.
• Select "Regression" and specify your independent (x) and dependent (y) variables.
• Review the outputs, which include regression coefficients, R-squared values, and p-values.

Excel simplifies the regression analysis process, making it accessible for students and professionals alike to understand the relationship between variables quickly and accurately.