Questions: A sports science group claims that due to improved training methods, professional cyclists burn a mean of less than 6300 calories during the annual Monaco Endurance Race. A study of 22 randomly selected professional cyclists finds that the sample mean number of calories the cyclists burn during the race is 6183 with a sample standard deviation of 302 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 0.05 level of significance, to support the claim that μ, the mean number of calories professional cyclists burn during the Monaco Endurance Race, is less than 6300. (a) State the null hypothesis H0 and the alternative hypothesis H1 that you would use for the test. H0 : H1 : (b) Perform a t test and find the p-value. Here is some information to help you with your t test. - The value of the test statistic is given by t=(x̄-μ)/(s/√n) - The p-value is the area under the curve to the left of the value of the test statistic.

A sports science group claims that due to improved training methods, professional cyclists burn a mean of less than 6300 calories during the annual Monaco Endurance Race. A study of 22 randomly selected professional cyclists finds that the sample mean number of calories the cyclists burn during the race is 6183 with a sample standard deviation of 302 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 0.05 level of significance, to support the claim that μ, the mean number of calories professional cyclists burn during the Monaco Endurance Race, is less than 6300. (a) State the null hypothesis H0 and the alternative hypothesis H1 that you would use for the test. H0 : H1 : (b) Perform a t test and find the p-value. Here is some information to help you with your t test. - The value of the test statistic is given by t=(x̄-μ)/(s/√n) - The p-value is the area under the curve to the left of the value of the test statistic.
Transcript text: A sports science group claims that due to improved training methods, professional cyclists burn a mean of less than 6300 calories during the annual Monaco Endurance Race. A study of 22 randomly selected professional cyclists finds that the sample mean number of calories the cyclists burn during the race is 6183 with a sample standard deviation of 302 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 0.05 level of significance, to support the claim that $\mu$, the mean number of calories professional cyclists burn during the Monaco Endurance Race, is less than 6300. (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$ that you would use for the test. $H_{0}$ : $\square$ $H_{1}$ : $\square$ (b) Perform a $t$ test and find the $p$-value. Here is some information to help you with your $t$ test. - The value of the test statistic is given by $t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$ - The $p$-value is the area under the curve to the left of the value of the test statistic.
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Solution

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Solution Steps

Step 1: State the Hypotheses

The null and alternative hypotheses for the test are defined as follows:

  • Null Hypothesis \( H_0: \mu \geq 6300 \)
  • Alternative Hypothesis \( H_1: \mu < 6300 \)
Step 2: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula:

\[ SE = \frac{s}{\sqrt{n}} = \frac{302}{\sqrt{22}} \approx 64.3866 \]

Step 3: Calculate the Test Statistic

The test statistic \( t \) is calculated using the formula:

\[ t_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{6183 - 6300}{64.3866} \approx -1.8171 \]

Step 4: Calculate the P-value

For a left-tailed test, the p-value is determined as follows:

\[ P = T(z) \approx 0.0417 \]

Step 5: Conclusion

At the significance level \( \alpha = 0.05 \), since the p-value \( 0.0417 < 0.05 \), we reject the null hypothesis \( H_0 \). This provides sufficient evidence to support the claim that the mean number of calories burned by professional cyclists during the Monaco Endurance Race is less than 6300.

Final Answer

\(\boxed{H_0 \text{ is rejected, supporting } H_1}\)

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