Two boats, the Prada (Italy) and the Oracle (USA), are competing for a spot in the upcoming America’s Cup race. They race over a part of the course several times. The sample times in minutes for the Prada were as follows: 12.9, 12.5, 11.0, 13.3, 11.2, 11.4, 11.6, 12.3, 14.2, and 11.3. The sample times in minutes for the Oracle were as follows: 14.1, 14.1, 14.2, 17.4, 15.8, 16.7, 16.1, 13.3, 13.4, 13.6, 10.8, and 19.0. For data analysis, the appropriate test is the t test: two-sample assuming unequal variances.
The next table shows the results of this independent t test. At the .05 significance level, can you conclude that there is a difference in their mean times? Explain these results to a person who knows about the t test for a single sample but who is unfamiliar with the t test for independent means.
No table.
To analyze whether there is a significant difference in the mean times of the Prada and Oracle boats, we can conduct an independent t test. This test allows us to compare the means of two independent groups and determine if there is a statistically significant difference between them.
First, we need to set up the null and alternative hypotheses. The null hypothesis assumes that there is no difference between the mean times of the Prada and Oracle boats, while the alternative hypothesis suggests that there is a difference between the two means.
Null Hypothesis (H0): There is no difference in the mean times of the Prada and Oracle boats.
Alternative Hypothesis (Ha): There is a difference in the mean times of the Prada and Oracle boats.
Next, we determine the level of significance, which is typically denoted as α (alpha). The question states that the significance level is .05, meaning we are willing to accept a 5% chance of making a Type I error (incorrectly rejecting the null hypothesis when it is true).
Now, let's perform the t test using the given sample times for each boat:
Prada sample times: 12.9, 12.5, 11.0, 13.3, 11.2, 11.4, 11.6, 12.3, 14.2, 11.3
Oracle sample times: 14.1, 14.1, 14.2, 17.4, 15.8, 16.7, 16.1, 13.3, 13.4, 13.6, 10.8, 19.0
We need to calculate the sample means and standard deviations for each group. Let's assume the Prada group is Sample 1 and the Oracle group is Sample 2.
Sample 1 statistics:
Sample mean (x̄1) = average of Prada sample times
Sample standard deviation (s1) = standard deviation of Prada sample times
Sample 2 statistics:
Sample mean (x̄2) = average of Oracle sample times
Sample standard deviation (s2) = standard deviation of Oracle sample times
Once we have these values, we can calculate the t statistic using the following formula:
t = (x̄1 - x̄2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where:
- x̄1 and x̄2 are the sample means
- s1 and s2 are the sample standard deviations
- n1 and n2 are the sample sizes
The degrees of freedom (df) can be calculated as:
df = (s1^2 / n1 + s2^2 / n2)^2 / [(s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1)]
Now, we can use the calculated t statistic and degrees of freedom to find the p-value from a t-table or using statistical software.
Finally, we compare the obtained p-value with the significance level (α = 0.05) to determine if we reject or fail to reject the null hypothesis.
If the p-value is less than α, we reject the null hypothesis and conclude that there is a significant difference in the mean times of the Prada and Oracle boats. If the p-value is greater than or equal to α, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a difference in the mean times.
Remember, the t test for independent means is used when comparing the means of two different groups, while the t test for a single sample is used to compare the mean of a single group to a known population mean.