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To determine if chocolate milk is as effective as other carbohydrate replacement drinks, nine male cyclists performed an intense workout followed by a drink and a rest period. At the end of the rest period, each cyclist performed an endurance trial in which he exercised until exhausted, and the time to exhaustion was measured. Each cyclist completed the entire regimen on two different days. On one day, the drink provided was chocolate milk, and on the other day the drink provided was a carbohydrate replacement drink. Data consistent with summary quantities in the paper "The Efficacy of Chocolate Milk as a Recovery Aid" (Medicine and Science in Sports and Exercise [2004]: S126) are given in the table at the bottom of the page. Is there evidence that the mean time to exhaustion is greater after chocolate milk than after a carbohydrate replacement drink? Use a significance level of \(\alpha=0.05\).

Step by step solution

01

State the hypotheses

The null hypothesis (\(H_0\)) states that there is no difference in the mean time to exhaustion between the two types of drinks: \(H_0: \mu_{chocolate} - \mu_{carbohydrate} = 0\) The alternative hypothesis (\(H_a\)) states that the mean time to exhaustion is greater after drinking chocolate milk: \(H_a: \mu_{chocolate} - \mu_{carbohydrate} > 0\)

02

Calculate the differences

Calculate the differences in time to exhaustion between the chocolate milk and carbohydrate replacement drink trials for each cyclist. This will give us a new set of data for the paired sample t-test.

03

Calculate the mean and standard deviation of the differences

Calculate the mean (\(\bar{d}\)) and standard deviation (\(s_d\)) of the differences found in Step 2.

04

Calculate the test statistic

Calculate the t-test statistic using the formula: \(t = \frac{\bar{d}}{s_d/\sqrt{n}}\) Where n is the number of cyclists.

05

Determine the critical value and make a decision

Using a t-distribution table with a significance level of \(\alpha=0.05\) and the appropriate degrees of freedom (\(n-1\)), find the critical value (\(t_{critical}\)). Compare the test statistic from Step 4 to the critical value. If the test statistic is greater than the critical value, we will reject the null hypothesis in favor of the alternative hypothesis, which means there is evidence that supports the mean time to exhaustion is greater after chocolate milk than after a carbohydrate replacement drink. If the test statistic is not greater than the critical value, we cannot reject the null hypothesis, and there is not enough evidence to conclude that the mean time to exhaustion is greater after chocolate milk.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a structured method used in statistics to determine if a certain premise holds for a data set. In this example, we're interested in understanding whether chocolate milk increases the time to exhaustion for cyclists when compared to a carbohydrate replacement drink. Normally, this process involves comparing a null hypothesis against an alternative hypothesis to draw a conclusion.
Here, we start by setting up the hypotheses:

• The null hypothesis (\( H_0 \)) suggests no difference between the effects of the two drinks on cyclists’ exhaustion times.
• The alternative hypothesis (\( H_a \)) proposes that chocolate milk leads to a greater time to exhaustion than the carbohydrate drink.

By comparing the test statistic to a critical value determined by our significance level, we can accept or reject the null hypothesis. This process of hypothesis testing allows us to make informed decisions based on statistical evidence.

Significance Level

The significance level, denoted by alpha (\( \alpha \)), plays a crucial role in hypothesis testing. It defines the probability of rejecting the null hypothesis when it is actually true. This is also known as a Type I error. In the context of this exercise, the researchers used a significance level of 0.05, meaning there’s a 5% risk of incorrectly concluding that chocolate milk has a greater effect.
Choosing an appropriate significance level is important:

• Higher significance levels (e.g., 0.10) increase the risk of Type I errors but make it easier to reject the null hypothesis.
• Lower significance levels (e.g., 0.01) reduce the risk but demand more stringent evidence to reject the null hypothesis.

Hence, this parameter helps balance the need for certainty with the desire to detect true differences.

Null Hypothesis

The null hypothesis (\( H_0 \)) is a starting assumption that there is no effect or difference between the groups being compared. It acts as the default position that researchers attempt to either reject or fail to reject based on their analysis. In this study, the null hypothesis claims there is no difference in the mean time to exhaustion after drinking chocolate milk versus a carbohydrate replacement drink.
When conducting a paired sample t-test like the one in this study, the null hypothesis specifically addresses the differences observed in paired observations. Here’s what you should remember about the null hypothesis:

• It's generally formulated to reflect "no effect" or "no change" scenarios.
• It provides a basis against which any observed data can be compared.

Overall, it serves as a tool for determining whether any observed effect is statistically significant or merely due to random variations.

Alternative Hypothesis

The alternative hypothesis (\( H_a \)) is what researchers hope to support by rejecting the null hypothesis. It posits that there is a real effect or difference present in the data. In our scenario with chocolate milk, the alternative hypothesis states that the mean time to exhaustion for cyclists is greater after consuming chocolate milk than after consuming a carbohydrate replacement.
This hypothesis gives direction to the research:

• It’s typically the opposite of the null hypothesis.
• In a one-sided test like this one, it specifies a direction of effect (greater, in our case).

An alternative hypothesis should be carefully formulated; it embodies the main prediction or insight that the analysis wishes to validate. Only by providing sufficient statistical evidence can the alternative hypothesis be supported, leading us to conclude that chocolate milk may indeed enhance endurance more effectively than its counterpart.