Question

The following data on gross efficiency (ratio of work accomplished per minute to calorie expenditure per minute) for trained endurance cyclists were given in the article "Cycling Efficiency Is Related to the Percentage of Type I Muscle Fibers" (Medicine and Science in Sports and Exercise [1992]: \(782-88\) ): \(\begin{array}{llllllll}18.3 & 18.9 & 19.0 & 20.9 & 21.4 & 20.5 & 20.1 & 20.1\end{array}\) \(\begin{array}{llllllll}20.8 & 20.5 & 19.9 & 20.5 & 20.6 & 22.1 & 21.9 & 21.2\end{array}\) \(\begin{array}{lll}20.5 & 22.6 & 22.6\end{array}\) a. Assuming that the distribution of gross energy in the population of all endurance cyclists is normal, give a point estimate of \(\mu\), the population mean gross efficiency. b. Making no assumptions about the shape of the population distribution, estimate the proportion of all such cyclists whose gross efficiency is at most 20 .

Short answer

The point estimate of the population mean (µ) can be found by adding all the data points and dividing by the total number of data points. To estimate the proportion of cyclists whose gross efficiency is at most 20, count the number of data points that are less than or equal to 20, then divide this number by the total number of data points.

Step-by-step solution

Calculate the Point Estimate of the Population Mean

Given the data points, add all of them and divide by the total number of data points. This will give the point estimate of the population mean (µ). \[ \mu = \frac{\sum_{i=1}^{n} x_i}{n} \] where \(x_i\) represents each data point and \(n\) is the total number of data points.

Calculate the Number of Cyclists with Efficiency at Most 20

To estimate the proportion of all cyclists whose gross efficiency is at most 20, we first need count the number of data points that are less than or equal to 20.

Estimate the Proportion

\[ Proportion = \frac{\text{Number of cyclists with efficiency at most 20}}{n} \] Divide the number of cyclists with gross efficiency at most 20 (from Step 2) by the total number of data points.