91Ó°ÊÓ

One type of experiment that might be performed by an exercise physiologist is as follows: Each person in a random sample is tested in a weight room to determine the heaviest weight with which he or she can perform an incline press five times with his or her dominant arm (defined as the hand that a person uses for writing). After a significant rest period, the same weight is determined for each individual's nondominant arm. The physiologist is interested in the differences in the weights pressed by each arm. The following data represent the maximum weights (in pounds) pressed by each arm for a random sample of 18 fifteen-year old girls. Assume that the differences in weights pressed by each arm for all fifteenyear old girls are approximately normally distributed. $$ \begin{array}{cccccc} \hline \text { Subject } & \begin{array}{c} \text { Dominant } \\ \text { Arm } \end{array} & \begin{array}{c} \text { Nondominant } \\ \text { Arm } \end{array} & \text { Subject } & \begin{array}{c} \text { Dominant } \\ \text { Arm } \end{array} & \begin{array}{c} \text { Nondominant } \\ \text { Arm } \end{array} \\ \hline 1 & 59 & 53 & 10 & 47 & 38 \\ 2 & 32 & 30 & 11 & 40 & 35 \\ 3 & 27 & 24 & 12 & 36 & 36 \\ 4 & 18 & 20 & 13 & 21 & 25 \\ 5 & 42 & 40 & 14 & 51 & 48 \\ 6 & 12 & 12 & 15 & 30 & 30 \\ 7 & 29 & 24 & 16 & 32 & 31 \\ 8 & 33 & 34 & 17 & 14 & 14 \\ 9 & 22 & 22 & 18 & 26 & 27 \\ \hline \end{array} $$ a. Make a \(99 \%\) confidence interval for the mean of the paired differences for the populations, where a paired difference is equal to the maximum weight for the dominant arm minus the maximum weight for the nondominant arm. b. Using the \(2 \%\) significance level, can you conclude that the average paired difference as defined in part a is positive?

Step by step solution

01

Computational Setup

First, calculate the differences between the weights lifted by the dominant and non-dominant arms for each individual. This creates a paired data set. Then compute the sample mean and the sample standard deviation of these differences.

02

Calculate the Confidence Interval

Use the sample mean and standard deviation to calculate the 99% confidence interval for the population mean difference. The formula to calculate the confidence interval is given by: \(\bar{x} ± Z_{\alpha/2}\frac{s}{sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size and \(Z_{\alpha/2}\) is the Z-score corresponding to the desired level of confidence.

03

Conduct the Hypothesis Test

With a 2% significance level, conduct a one-sample t-test on the differences to see if the population mean difference is greater than 0, i.e, if the dominant hand lifts more weight on average than the non-dominant hand. The null hypothesis is that the mean difference is 0, while the alternative hypothesis is that the mean difference is greater than 0.

04

Interpret the Results

Based on the results of the confidence interval and hypothesis test, answer the questions posed in part a and b. If the entire confidence interval from step 2 lies above 0, you can conclude at a 99% level of confidence that the dominant hand lifts more weight than the non-dominant hand on average. If the t-test results in a p-value less than 0.02, you can reject the null hypothesis at a 2% significance level and conclude that the dominant hand lifts more weight than the non-dominant hand on average.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Confidence Interval

A confidence interval is a statistical tool used to estimate the range within which a population parameter is likely to fall. In this exercise, we aim to calculate a 99% confidence interval for the difference in maximum weights lifted by the dominant and nondominant arms. This involves:

• Calculating the sample mean difference, which is the average of all individual differences.
• Determining the sample standard deviation, which provides a measure of the dispersion of these differences.
• Using the standard normal distribution to find the Z-score for our confidence level.

This Z-score is then used along with the mean and standard deviation to calculate the confidence interval. The formula for the confidence interval is: \[ \bar{x} \pm Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \] where \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, \( n \) is the sample size, and \( Z_{\alpha/2} \) is the Z-score corresponding to our 99% confidence level. This interval provides us with a range that, with 99% certainty, contains the true mean difference between the weights lifted by the two arms.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this situation, we use a one-sample t-test to analyze the mean difference in weights lifted by dominant versus nondominant arms. There are two hypotheses to consider:

• Null Hypothesis (\(H_0\)): The mean difference is 0, suggesting no difference in strength between the two arms.
• Alternative Hypothesis (\(H_a\)): The mean difference is greater than 0, suggesting the dominant arm is stronger.

The test statistic for a one-sample t-test is computed using the sample mean, standard deviation, and size. We compare this statistic to the critical value from the t-distribution at the given significance level. If the statistic exceeds this critical value, we reject the null hypothesis, concluding that there is a significant mean difference.

Significance Level

The significance level (\(\alpha\)) is a critical component of hypothesis testing. It defines the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. In part (b) of the exercise, we use a significance level of 2%, meaning there is a 2% chance of concluding that the dominant arm is stronger if this is not actually the case. When conducting a hypothesis test:

• We determine the critical value from the t-distribution that corresponds to this significance level.
• We compare the computed test statistic to this critical value.

This process helps us control the likelihood of making incorrect decisions and ensures that our conclusions are reliable at the specified confidence level.

Normal Distribution

The normal distribution is a key concept in statistics, often used because many datasets naturally follow a bell-shaped pattern. In this exercise, it's assumed that differences in weights lifted by the dominant and nondominant arms are normally distributed for all fifteen-year-old girls. This assumption is crucial for several reasons:

• It allows us to use the Z-score in constructing the confidence interval for the mean difference.
• It justifies the use of the t-test in hypothesis testing.

The normal distribution is characterized by its symmetry about the mean, with data tapering off equally on both sides. This symmetry ensures that statistical inferences, such as confidence intervals and tests, are valid under the normality assumption. Without this assumption, the reliability of our conclusions may be compromised.