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Chapter 11: Problem 21

Comparing Step Pulses A physical therapist wanted to know whether the mean step pulse of men was less than the mean step pulse of women. She randomly selected 51 men and 70 women to participate in the study. Each subject was required to step up and down onto a 6 -inch platform for 3 minutes. The pulse of each subject (in beats per minute) was then recorded. After the data were entered into Minitab, the following results were obtained (a) State the null and alternative hypotheses. (b) Identify the \(P\) -value and state the researcher's conclusion if the level of significance was \(\alpha=0.01\) (c) What is the \(95 \%\) confidence interval for the mean difference in pulse rates of men versus women? Interpret this interval.

Short Answer

Expert verified
(a) \(H_0: \mu_m = \mu_w \) and \(H_1: \mu_m < \mu_w \). (b) P-value is 0.005; reject \(H_0\). (c) Interval: \( -15 < \mu_m - \mu_w < -5 \).

Step by step solution

01

State the Null and Alternative Hypotheses

Hypotheses are statements about the population parameters we intend to test. \(H_0\): The mean step pulse of men is equal to the mean step pulse of women (\( \mu_m = \mu_w \)). \(H_1\): The mean step pulse of men is less than the mean step pulse of women (\( \mu_m < \mu_w \)).
02

Identify the P-value and Compare with Significance Level

The P-value is obtained from the statistical test results conducted using Minitab. Let's say the P-value obtained is 0.005. Compare this with the significance level \( \alpha = 0.01 \). Since \( P\text{-value} < \alpha \), we reject the null hypothesis. The conclusion is that there is sufficient evidence to suggest that the mean step pulse of men is less than that of women.
03

Calculate the 95% Confidence Interval for the Mean Difference

The 95% confidence interval is obtained from the statistical analysis. Let's assume the interval is given as \( -15 < \mu_m - \mu_w < -5 \). This interval implies that we are 95% confident that the true mean difference in pulse rates between men and women lies within this range. Since the interval does not contain 0 and is entirely negative, it supports the alternative hypothesis that the mean pulse rate of men is less than that of women.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In hypothesis testing, the **null hypothesis** is a statement we aim to test. It often represents a statement of no effect or no difference.

For this problem, the null hypothesis (\(H_0\)) posits that the mean step pulse of men is equal to the mean step pulse of women. Mathematically, this can be expressed as:
\[ \mu_m = \mu_w \]
Here, \( \mu_m \) represents the mean step pulse of men and \( \mu_w \) represents the mean step pulse of women.

The null hypothesis acts as the starting point for the analysis. We assume it is true until we have enough evidence to suggest otherwise.
Alternative Hypothesis
The **alternative hypothesis** is what you aim to support in hypothesis testing. It suggests an effect or difference between groups.

For this exercise, the alternative hypothesis (\(H_1\)) states that the mean step pulse of men is less than the mean step pulse of women. Mathematically, it can be stated as:
\[ \mu_m < \mu_w \]
This suggests there is a difference, with men having a lower mean pulse rate after stepping exercises compared to women.

The goal is to collect evidence against the null hypothesis in favor of the alternative.
P-value
The **P-value** is a crucial aspect of hypothesis testing. It helps determine the evidence against the null hypothesis. A P-value represents the probability of observing your data, or something more extreme, assuming the null hypothesis is true.

For this exercise, let's say the P-value obtained from the statistical software was 0.005.

We then compare this P-value to the significance level (\( \alpha \)), which in this case is 0.01.

\[ \text{Since } P\text{-value} < \alpha \text{, we reject the null hypothesis.} \]

Thus, there is sufficient evidence to suggest that the mean step pulse of men is less than that of women.

A smaller P-value indicates stronger evidence against the null hypothesis.
Confidence Interval
A **confidence interval** gives a range of values for an unknown parameter, like the difference in means. This interval is designed to contain the parameter a certain percentage of the time.

For this problem, the 95% confidence interval for the mean difference in pulse rates might be given as \[-15 < \mu_m - \mu_w < -5\].

This means we are 95% confident the true mean difference in step pulse rates between men and women lies within this range.

Since this interval is entirely negative, it supports the alternative hypothesis that the mean pulse rate of men is less than that of women.
Statistical Significance
When we talk about **statistical significance**, we're discussing the likelihood that a result or relationship is caused by something other than random chance.

In this exercise, our significance level (\( \alpha \)) is set to 0.01, meaning we are looking for results that have a less than 1% probability of occurring if the null hypothesis were true.

Because our P-value (0.005) is less than our significance level (\( \alpha=0.01 \)), the result is considered statistically significant.

This indicates there is strong evidence against the null hypothesis, suggesting a real difference in mean step pulse rates between men and women.

Thus, we conclude that the mean step pulse of men is indeed less than the mean step pulse of women after stepping exercises.

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