Questions: A company is developing a new high performance wax for cross country ski racing. In order to justify the price marketing wants, the wax needs to be very fast. Specifically, the mean time to finish their standard test course should be less than 55 seconds for a former Olympic champion. To test it, the champion will ski the course 8 times. The champion's times (selected at random) are 53.2, 64.1, 51.5, 52.8, 46.8, 46.4, 52.2, and 43.7 seconds to complete the test course. Should they market the wax? Assume the assumptions and conditions for appropriate hypothesis testing are met for the sample. Use 0.05 as the P-value cutoff level. Choose the correct null and alternative hypotheses below. A. H0: μ>55 B. H0: μ=55 HA · μ=55 HA · μ>55 C. H0: μ=55 D. H0: μ<55 HA · μ<55 HA · μ=55 Calculate the test statistic. t= (Round to three decimal places as needed.)

A company is developing a new high performance wax for cross country ski racing. In order to justify the price marketing wants, the wax needs to be very fast. Specifically, the mean time to finish their standard test course should be less than 55 seconds for a former Olympic champion. To test it, the champion will ski the course 8 times. The champion's times (selected at random) are 53.2, 64.1, 51.5, 52.8, 46.8, 46.4, 52.2, and 43.7 seconds to complete the test course. Should they market the wax? Assume the assumptions and conditions for appropriate hypothesis testing are met for the sample. Use 0.05 as the P-value cutoff level. Choose the correct null and alternative hypotheses below. A. H0: μ>55 B. H0: μ=55 HA · μ=55 HA · μ>55 C. H0: μ=55 D. H0: μ<55 HA · μ<55 HA · μ=55

Calculate the test statistic. t= (Round to three decimal places as needed.)
Transcript text: A company is developing a new high performance wax for cross country ski racing. In order to justify the price marketing wants, the wax needs to be very fast. Specifically, the mean time to finish their standard test course should be less than 55 seconds for a former Olympic champion. To test it, the champion will ski the course 8 times. The champion's times (selected at random) are 53.2, 64.1, $51.5,52.8,46.8,46.4,52.2$, and 43.7 seconds to complete the test course. Should they market the wax? Assume the assumptions and conditions for appropriate hypothesis testing are met for the sample. Use 0.05 as the P-value cutoff level. Choose the correct null and alternative hypotheses below. A. $H_{0}: \mu>55$ B. $H_{0}: \mu=55$ $H_{A} \cdot \mu=55$ $H_{A} \cdot \mu>55$ C. $H_{0}: \mu=55$ D. $H_{0}: \mu<55$ $H_{A} \cdot \mu<55$ $H_{A} \cdot \mu=55$ Calculate the test statistic. $\mathrm{t}=$ $\square$ (Round to three decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Sample Mean

The sample mean \( \bar{x} \) is calculated as follows:

\[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{410.7}{8} = 51.337 \]

Step 2: Calculate the Sample Variance and Standard Deviation

The sample variance \( s^2 \) is given by:

\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 38.937 \]

The sample standard deviation \( s \) is then:

\[ s = \sqrt{s^2} = \sqrt{38.937} = 6.24 \]

Step 3: Calculate the Standard Error

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

\[ SE = \frac{s}{\sqrt{n}} = \frac{6.24}{\sqrt{8}} = 2.206 \]

Step 4: Calculate the Test Statistic

The test statistic \( t \) for the left-tailed test is calculated as:

\[ t = \frac{\bar{x} - \mu_0}{SE} = \frac{51.337 - 55}{2.206} = -1.66 \]

Step 5: Calculate the P-value

For a left-tailed test, the P-value is determined to be:

\[ P = T(z) = 0.07 \]

Step 6: Decision Based on P-value

Given the significance level \( \alpha = 0.05 \):

Since \( P = 0.07 > 0.05 \), we fail to reject the null hypothesis.

Final Answer

The null hypothesis states that the mean time to finish the course is equal to or greater than 55 seconds. Since we fail to reject the null hypothesis, we conclude that the wax should not be marketed.

\(\boxed{\text{Do not market the wax.}}\)

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