Two boats, the Prada (Italy) and the Oracle (U.S.A.), are competing for a spot in the upcoming America’s Cup race. They race over a part of the course several times. Below are the sample times in minutes. Assume the population standard deviations are not the same. At the .05 significance level, can we conclude that there is a difference in their mean times?
We do not have access to your sample times.
To determine if there is a significant difference in the mean times of the Prada and Oracle boats, we can use a two-sample t-test. This test compares the means of two independent groups to determine if the difference between them is statistically significant.
Here's the step-by-step process to conduct the hypothesis test:
Step 1: State the null and alternative hypothesis:
The null hypothesis (H0) assumes that there is no difference between the mean times of the two boats:
H0: μ1 = μ2
The alternative hypothesis (Ha) assumes that there is a difference between the mean times of the two boats:
Ha: μ1 ≠ μ2
Step 2: Determine the significance level:
The significance level (α) is given as 0.05 in this case.
Step 3: Identify the test statistic:
For comparing the means of two independent groups, we can use the two-sample t-test.
Step 4: Compute the test statistic:
Calculate the t-test statistic using the sample data provided. This can be done using statistical software or by hand calculations:
Step 5: Determine the critical value:
The critical value(s) depend on the significance level (α) and the degrees of freedom (df) associated with the t-distribution. The degrees of freedom can be calculated as follows:
df = n1 + n2 - 2
Step 6: Compare the test statistic with the critical value:
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference in the mean times of the two boats. If the absolute value of the test statistic is less than the critical value, we fail to reject the null hypothesis.
Step 7: Interpret the result:
Based on the comparison in Step 6, we can conclude whether there is evidence to support the claim of a difference in mean times between the Prada and Oracle boats.
By following these steps and conducting the appropriate calculations, you can determine if there is a significant difference in mean times between the Prada and Oracle boats at the 0.05 significance level.