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A sports science group claims that due to improved training methods, professional cyclists burn a mean of less than 6800 calories during the annual Monaco Endurance Race. (This would be an improvement on the previously accepted value of 6800 calories.) A study of 18 randomly selected professional cyclists finds that the sample mean number of calories the cyclists burn during the race is 6623 with a sample standard deviation of 364 calories. Assume that the population of numbers of calories burned by professional cyclists during the race is approximately normally distributed.

Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the a=0.05 level of significance, to support the claim that u, the mean number of calories professional cyclists burn during the Monaco Endurance Race, is less than 6800.

(a) State the null hypothesis Hv0 and the alternative hypothesis Hv1 that you would use for the test.

5 answers

To perform the hypothesis test in this scenario, we will define the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)) as follows:

Null Hypothesis (\(H_0\)): The mean number of calories burned by professional cyclists during the Monaco Endurance Race is equal to 6800 calories.
\(H_0: \mu = 6800\)

Alternative Hypothesis (\(H_1\)): The mean number of calories burned by professional cyclists during the Monaco Endurance Race is less than 6800 calories.
\(H_1: \mu < 6800\)

In this case, we are interested in determining if the new training methods have resulted in a mean calorie burn that is less than the previously accepted value of 6800 calories.

What is the number of degrees of freedom?
Is it one or two tailed?
what is the test statistic (rounded to 3 decimal places)
what is the p-value (rounded to 3 decimal places)

To conduct the hypothesis test, we will calculate the degrees of freedom, determine whether the test is one-tailed or two-tailed, compute the test statistic, and find the p-value step-by-step.

  1. Degrees of Freedom: The degrees of freedom for a t-test is given by: \[ df = n - 1 \] where \(n\) is the sample size. In this case, \(n = 18\): \[ df = 18 - 1 = 17 \]

  2. One-tailed or Two-tailed: Since we are testing whether the mean is less than 6800 (i.e., we have a specific direction), this is a one-tailed test.

  3. Test Statistic: We will compute the t-test statistic using the following formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where: \(\bar{x} = 6623\) (sample mean),
    \(\mu_0 = 6800\) (hypothesized population mean),
    \(s = 364\) (sample standard deviation),
    \(n = 18\) (sample size).

    Now plug in the values: \[ t = \frac{6623 - 6800}{364 / \sqrt{18}} \] First, calculate the denominator: \[ s / \sqrt{n} = 364 / \sqrt{18} \approx 364 / 4.243 \approx 85.8 \] Then calculate the t-statistic: \[ t = \frac{-177}{85.8} \approx -2.063 \] Therefore, rounded to three decimal places, the test statistic is: \[ t \approx -2.063 \]

  4. P-value: To find the p-value for the one-tailed test, we can use a t-distribution table or a calculator to find the probability of observing a t statistic of -2.063 with 17 degrees of freedom.

    Using a t-distribution calculator or software (like Python, R, or statistical calculators), we can find: \[ p\text{-value} \approx 0.027 \] Therefore, the p-value rounded to three decimal places is: \[ p\text{-value} \approx 0.027 \]

Summary:

  • Degrees of Freedom: 17
  • One-tailed or Two-tailed: One-tailed
  • Test Statistic: -2.063
  • P-value: 0.027
(c)Based on your answer to part (b), choose what can be concluded, at the 0.05 level of significance, about the claim made by the sports science group.

1) Since the p-value is less than (or equal to) the level of significance, the null hypothesis is rejected. So, there is enough evidence to support the claim that the mean number of calories professional cyclists burn during the Monaco Endurance Race is less than 6800.

2) Since the p-value is less than (or equal to) the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to support the claim that the mean number of calories professional cyclists burn during the Monaco Endurance Race is less than 6800.

3) Since the p-value is greater than the level of significance, the null hypothesis is rejected. So, there is enough evidence to support the claim that the mean number of calories professional cyclists burn during the Monaco Endurance Race is less than 6800.

or 4) Since the p-value is greater than the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to support the claim that the mean number of calories professional cyclists burn during the Monaco Endurance Race is less than 6800.

Based on the results from part (b), we found that the p-value is approximately 0.027, and we are testing at a 0.05 level of significance.

Since the p-value (0.027) is less than the level of significance (0.05), we reject the null hypothesis. Therefore, we have enough evidence to support the claim made by the sports science group that the mean number of calories professional cyclists burn during the Monaco Endurance Race is less than 6800 calories.

The correct conclusion based on this information is:

1) Since the p-value is less than (or equal to) the level of significance, the null hypothesis is rejected. So, there is enough evidence to support the claim that the mean number of calories professional cyclists burn during the Monaco Endurance Race is less than 6800.