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Introductory statistics students fill out a survey on the first day of class. One of the questions asked is "How many hours of exercise do you typically get each week?" Responses for a sample of 50 students are introduced in Example 3.25 on page 244 and stored in the file ExerciseHours. The summary statistics are shown in the computer output below. The mean hours of exercise for the combined sample of 50 students is 10.6 hours per week and the standard deviation is 8.04 . We are interested in whether these sample data provide evidence that the mean number of hours of exercise per week is different between male and female statistics students. $$\begin{array}{lllrrrr} \text { Variable } & \text { Gender } & \text { N } & \text { Mean } & \text { StDev } & \text { Minimum } & \text{ Maximum } \\\\\text { Exercise } & \mathrm{F} & 30 & 9.40 & 7.41 & 0.00 & 34.00 \\\& \mathrm{M} & 20 & 12.40 & 8.80 & 2.00 & 30.00\end{array}$$ Discuss whether or not the methods described below would be appropriate ways to generate randomization samples that are consistent with \(H_{0}: \mu_{F}=\mu_{M}\) vs \(H_{a}: \mu_{F} \neq \mu_{M} .\) Explain your reasoning in each case. (a) Randomly label 30 of the actual exercise values with "F" for the female group and the remaining 20 exercise values with "M" for the males. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\). (b) Add 1.2 to every female exercise value to give a new mean of 10.6 and subtract 1.8 from each male exercise value to move their mean to 10.6 (and match the females). Sample 30 values (with replacement) from the shifted female values and 20 values (with replacement) from the shifted male values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) (c) Combine all 50 sample values into one set of data having a mean amount of 10.6 hours. Select 30 values (with replacement) to represent a sample of female exercise hours and 20 values (also with replacement) for a sample of male exercise values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) .

Step by step solution

01

Understanding Data and Formulating Hypotheses

Evaluate the data provided for the exercise hours of female and male students. The null hypothesis is that the mean hours of exercise for both genders are equal (\(H_0: \mu_{F}=\mu_{M}\)). The alternative hypothesis states that they are not equal (\(H_a: \mu_{F} \neq \mu_{M}\)).

02

Evaluating Method A

Assess the appropriateness of Method A, which suggests randomly labelling 30 of the exercise values as female and remaining 20 as male. This method is not appropriate because it ignores the actual genders of the students. We must maintain the structure of the original gender assignments when generating randomization samples under the null hypothesis.

03

Evaluating Method B

Assess the appropriateness of Method B, which involves adjusting the mean hours for both genders to match the overall average, then re-sampling. This method is not appropriate as it artificially standardizes the means, which is not what we expect under the null hypothesis. Our goal under the null hypothesis is to adjust the labels of the observations, not the means of the groups.

04

Evaluating Method C

Evaluate the appropriateness of Method C, which combines all exercise hours, randomly select 30 for females, and 20 for males. This method is appropriate because under the null hypothesis, we expect no difference in exercise time due to gender. By amalgamating the data into one group and randomly assigning observations, we maintain the original structure of the data while respecting the presumption of the null hypothesis.

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Key Concepts

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

Introductory Statistics

Introductory statistics is a branch of mathematics that helps us understand and interpret data. It's great for beginners because it covers essential topics like data collection, sampling methods, and basic statistical calculations. These tools are crucial when we're trying to understand how different variables, like gender, might impact other variables, such as exercise frequency.

At its core, introductory statistics provides the foundation for more advanced analysis by teaching concepts such as:

• Descriptive statistics: This involves summarizing and describing data using measures like mean, median, mode, and standard deviation.
• Inferential statistics: Here, statistical methods are used to make predictions or inferences about a population based on a sample from that population. This is particularly useful in hypothesis testing.

Understanding these basics helps us formulate and test theories about the world, like whether there are gender differences in exercise habits.

Hypothesis Testing

Hypothesis testing is a procedure used in statistics to determine if there is enough evidence to support a specific claim about a population. This process involves several key steps:

• Formulate hypotheses: We start by setting up a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)). In our exercise, the null hypothesis suggests there is no difference in exercise hours between genders (\(H_0: \mu_{F}=\mu_{M}\)), while the alternative hypothesis indicates there is a difference (\(H_a: \mu_{F} eq \mu_{M}\)).
• Collect data and calculate statistics: For the hypothesis test, we collect data (as given in the exercise) and compute relevant statistics such as means and standard deviations.
• Determine a test method: We select an appropriate test to use based on our data and hypotheses. In our example, methods like resampling or randomization tests are considered.
• Make a decision: By comparing the observed data with what we'd expect if the null hypothesis were true, we decide whether to reject or not reject the null hypothesis. The decision is often based on a p-value, which indicates the probability of observing our data if the null hypothesis is true.

Hypothesis testing allows us to make informed decisions and move beyond simple observations, adding rigor and credibility to our analyses.

Gender Differences in Exercise

Exploring gender differences in exercise is an intriguing area of study in statistics and can provide valuable insights into social and health-related behaviors. By analyzing data on exercise habits, we can uncover whether these habits differ significantly between males and females and identify trends that could inform policies or health interventions.

When examining such differences, we look at:

• Data collection methods: Ensuring that data collection is fair and unbiased is crucial. This means gathering information uniformly across groups.
• Summary statistics: Looking at means, medians, and ranges gives us an initial view of any potential differences. For instance, in the exercise data provided, we see distinct averages for each gender.
• Statistical tests: Using methods like hypothesis tests allows us to examine whether observed differences are significant or if they could have occurred by chance.

Understanding gender differences in exercise can guide tailored health programs, promote equal opportunities for participation in physical activities, and lead to healthier lifestyles for all groups.