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The paper "Playing Active Video Games Increases Energy Expenditure in Children" (Pediatrics [2009]: \(534-539\) ) describes an interesting investigation of the possible cardiovascular benefits of active video games. Mean heart rate for healthy boys age 10 to 13 after walking on a treadmill at \(2.6 \mathrm{~km} /\) hour for 6 minutes is 98 beats per minute (bpm). For each of 14 boys, heart rate was measured after 15 minutes of playing Wii Bowling. The resulting sample mean and standard deviation were 101 bpm and 15 bpm, respectively. For purposes of this exercise, assume that it is reasonable to regard the sample of boys as representative of boys age 10 to 13 and that the distribution of heart rates after 15 minutes of Wii Bowling is approximately normal. a. Does the sample provide convincing evidence that the mean heart rate after 15 minutes of Wii Bowling is different from the known mean heart rate after 6 minutes walking on the treadmill? Carry out a hypothesis test using \(\alpha=.01\). b. The known resting mean heart rate for boys in this age group is \(66 \mathrm{bpm}\). Is there convincing evidence that the mean heart rate after Wii Bowling for 15 minutes is higher than the known mean resting heart rate for boys of this age? Use \(\alpha=.01\). c. Based on the outcomes of the tests in Parts (a) and (b), write a paragraph comparing the benefits of treadmill walking and Wii Bowling in terms of raising heart rate over the resting heart rate.

Step by step solution

01

Setting up the Hypothesis for Treadmill vs Wii Bowling

Setting the null hypothesis \(H_0: \mu = 98\), the mean heart rate after Wii Bowling is equal to the mean heart rate after treadmill. The alternative hypothesis \(H_a: \mu \neq 98\), the mean heart rate after Wii Bowling is not equal to the mean heart rate after treadmill.

02

Conducting the t-test for Treadmill vs Wii Bowling

We use a one-sample t-test to determine whether there is a significant difference. Using the given sample mean = 101 bpm, standard deviation = 15 bpm, and sample size = 14. The formula for the t-value is \( t = \frac{(\bar{x}-\mu_0)}{(s/\sqrt{n})} \). Plug and calculate to get the t-value.

03

Finding p-value and conclusion for Treadmill vs Wii Bowling

Find the p-value using the t-distribution. If the p-value <= \(\alpha = .01\), then reject the null hypothesis. Compare and draw conclusions.

04

Setting up the Hypothesis for Resting Heart Rate vs Wii Bowling

Similarly, formulate the null hypothesis \(H_0: \mu = 66\), the mean heart rate after Wii Bowling is equal to the resting heart rate. And the alternative hypothesis \(H_a: \mu > 66\), the mean heart rate after Wii Bowling is more than the resting heart rate.

05

Conducting the t-test for Resting Heart Rate vs Wii Bowling

Implement the one-sample t-test again with the relevant values. The sample std, mean and size remain the same, whereas mu changes to 66. Calculate the t-value using the t-test formula.

06

Finding p-value and conclusion for Resting Heart Rate vs Wii Bowling

Evaluate the p-value based on the t-value. Here, as we're testing whether the Wii Bowling heart rate is 'greater' than the resting heart rate, this is a one-tailed test. If p-value split in half is <= to \(\alpha = 0.01\), then reject the null hypothesis.

07

Comparative Analysis

Use the conclusions from hypothesis tests to compare the benefits of Wii Bowling and Treadmill in terms of raising heart rate over the resting heart rate and describe in a paragraph.

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Key Concepts

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

One-Sample T-Test

The one-sample t-test is a statistical tool used to determine whether there is a significant difference between the mean of a single sample and a known mean (population mean). It's often applied in scenarios where the population variance is unknown and the sample size is relatively small (typically less than 30).

For instance, in the case from the exercise involving mean heart rates post treadmill walking compared to Wii Bowling, a one-sample t-test helps in understanding whether playing Wii Bowling significantly affects heart rate compared to walking on a treadmill. This is executed by formulating two hypotheses - the null hypothesis, which posits no effect (no significant difference in heart rates), and the alternative hypothesis, which proposes that there is an effect (a significant difference in heart rates).

The test statistic is computed using the formula: \( t = \frac{(\bar{x}-\mu_0)}{(s/\sqrt{n})} \), where \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, \( n \) is the sample size, and \( \mu_0 \) is the known population mean. This statistic is then compared against a t-distribution to derive the p-value, which is used to infer statistical significance.

P-Value Calculation

The p-value is a critical concept in hypothesis testing that measures the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis is correct. The smaller the p-value, the stronger the evidence against the null hypothesis.

Mathematically, the p-value is calculated based on the test statistic (such as the t-value from the one-sample t-test) and the appropriate distribution. In the exercise regarding heart rate analysis, p-values are derived from a t-distribution, reflecting the degrees of freedom from our sample data.

The choice of the significance level \( \alpha \), which is commonly set at 0.05 or 0.01, serves as a threshold. If the calculated p-value is less than or equal to \( \alpha \), the null hypothesis is rejected in favor of the alternative hypothesis. Specifically, in parts (a) and (b) of our exercise with \( \alpha = 0.01 \), heart rate data are assessed to discover if the mean rates after Wii Bowling differs from either the treadmill mean rate or the resting heart rate.

Heart Rate Statistical Analysis

Heart rate statistical analysis involves using various statistical methods to evaluate heart rate data. In the given exercise, we're comparing the effects of different activities on heart rate by employing the one-sample t-test. This test helps to investigate if certain activities, like Wii Bowling, can significantly increase the heart rate compared to a baseline (treadmill walking or resting rate) in the context of cardiovascular health for a specific age group.

In part (a), we explore if Wii Bowling differs from treadmill walking, while in part (b), we examine if Wii Bowling elevates heart rate more than the resting rate. Here, the analysis also distinguishes between two-tailed and one-tailed tests, relevant when the direction of the effect matters. For example, asserting that Wii Bowling increases (not just changes) heart rate is a one-tailed hypothesis.

Finally, the comparison of the benefits between treadmill walking and Wii Bowling, regarding their effectiveness at raising heart rates above resting level, combines the insights obtained from the statistical tests, offering conclusions about their relative cardiovascular benefits for the demographic in question.