Question

The manufacturer of exercise machines for fitness centers has designed two new

elliptical machines that are meant to increase cardiovascular fitness. The two machines

are being tested on $30$volunteers at a fitness center near the company’s headquarters.

The volunteers are randomly assigned to one of the machines and use it daily for two

months. A measure of cardiovascular fitness is administered at the start of the

experiment and again at the end. The following stem plot contains the differences (After

– Before) in the two scores for the two machines. Note that higher scores indicate larger

gains in fitness.

a. Write a few sentences comparing the distributions of cardiovascular fitness gains

from the two elliptical machines.

b. Which machine should be chosen if the company wants to advertise it as achieving

the highest overall gain in cardiovascular fitness? Explain your reasoning.

c. Which machine should be chosen if the company wants to advertise it as achieving

the most consistent gain in cardiovascular fitness? Explain your reasoning.

d. Give one reason why the advertising claims of the company (the scope of inference)

for this experiment would be limited. Explain how the company could broaden that

scope of inference.

Short answer

(a) Machine A is symmetric and machine B is skewed.

(b) Machine B should be choose.

(c) Machine A should be choose.

(d) Experiment was conducted only in one fitness center.

Step-by-step solution

Part (a) Step 1: Given Information

We need to compare the elliptical machines.

Part (a) Step 2: Explanation

Machine A is essentially symmetric because the majority of the data values are located in the stem and leaf diagram, but machine B is left skewed because the most of the data values are located in the bottom of the stem and leaf diagram. Because most of the data values are centred at stem $4$, machine B has a somewhat higher Center than machine A. Because machine A only uses four stems, machine B has a wider spread than machine A.

Part (b) Step 1: Given Information

We need to find which machine have highest overall gain.

Part (b) Step 2: Explanation

This data set has the most value, and it is the set that should be used for advertising.

As a result, machine B has a large overall gain.

Part (c) Step 1: Given Information

We need to find that which should be chosen for consistent gain .

Part (c) Step 2: Explanation

Machine A has a smaller spread because it has no value in the beginning and end rows. As a result, machine A has the most constant gain.

Part (d) Step 1: Given Information

We need to tell the scope of inference.

Part (d) Step 2: Explanation

This study relied on volunteers, and these volunteers may differ in some ways from the general public who are interested in cardiovascular fitness.