Answer:
Based on the given race times, Bjork Larsen should not buy the new racing wax for cross-country skis if his criterion is to average less than 55 seconds on the course.
Step-by-step explanation:
To determine whether Bjork Larsen should buy the new racing wax for cross-country skis, we can perform a hypothesis test.
The null hypothesis (H0) would be that the average race time is equal to or greater than 55 seconds, while the alternative hypothesis (Ha) would be that the average race time is less than 55 seconds.
Let's calculate the average race time and perform the hypothesis test using a significance level of 0.05.
The 8 race times are: 56.3, 65.9, 50.5, 52.4, 46.5, 57.8, 52.2, and 43.2 seconds.
Step 1: Calculate the sample mean
The sample mean (x) is calculated by summing up all the race times and dividing by the number of races (8 in this case).
x = (56.3 + 65.9 + 50.5 + 52.4 + 46.5 + 57.8 + 52.2 + 43.2) / 8 = 51.95 seconds
Step 2: Calculate the sample standard deviation
The sample standard deviation (s) measures the variability of the race times.
s = √[ (56.3 - 51.95)² + (65.9 - 51.95)² + (50.5 - 51.95)² + (52.4 - 51.95)² + (46.5 - 51.95)² + (57.8 - 51.95)² + (52.2 - 51.95)² + (43.2 - 51.95)² ] / (8 - 1)
s ≈ 7.32 seconds
Step 3: Perform the hypothesis test
We can perform a one-sample t-test to compare the sample mean to the hypothesized mean of 55 seconds.
The t-value is calculated as: t = (x - μ) / (s / √n)
Where:
x is the sample mean
μ is the hypothesized mean (55 seconds)
s is the sample standard deviation
n is the sample size (8)
t = (51.95 - 55) / (7.32 / √8)
≈ -1.49
Using a t-table or statistical software, we can find the critical t-value for a one-tailed test with a significance level of 0.05 and 7 degrees of freedom (n - 1 = 8 - 1 = 7).
The critical t-value is approximately -1.895.
Since the calculated t-value (-1.49) is not less than the critical t-value (-1.895), we fail to reject the null hypothesis.
This means that there is not enough evidence to conclude that the average race time is less than 55 seconds.
Therefore,
Based on the given race times, Bjork Larsen should not buy the new racing wax for cross-country skis if his criterion is to average less than 55 seconds on the course.