Two Boats the Picadilly and the Orange are competing for a spot in the Cup races. They race over a part of the course SEVERAL TIMES. The sample times in minutes for the Picadilly are 12.9, 12.5, 11.0, 13.3, 11.2, 11.4, 11.6, 12.3, 14.2, and 11.3. The sample times for the Orange are, 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 sampling assuming unequal variances. The next table shows the resuklts of the independent T-TEST. At the .05 significance level, can we conclude that there is a difference in their mean times? EXPLAIN THESE RESULTS to a person who knows about the t test for a singel sample but is unfamiliar with the t test for independent means.
pICADILLY
12.170 MEAN
1.056 SD
10 N
ORANGE
14.875 MEAN
2.208 SD
12 N
16 df
-2.7050 diff (PICA-ORANGE)
0.7196 s err of diff
0 hypoth diff
- 3.76 t
.0017 p value two tailed
-4.2304 C I 95% lower
-1.1796 C I 95% UPPER
1.5254 M.O.E
the next table shows results of this independent t-test. At 0.05significant level can you conclude that there is a difference in their mean times? explain these results to a person who knows about t-test for a single sample but who is unfamiliar with the t-test for independent means
To determine if there is a significant difference in the mean times of the Picadilly and Orange boats, a t-test for independent means was conducted. The t-test allows us to compare the means of two independent groups to see if they are statistically different.
Here are the key results of the t-test:
1. Sample Statistics:
- Picadilly: The mean time is 12.170 minutes, with a standard deviation of 1.056 minutes, based on a sample size of 10.
- Orange: The mean time is 14.875 minutes, with a standard deviation of 2.208 minutes, based on a sample size of 12.
2. Hypotheses:
- Null hypothesis (H0): There is no difference in the mean times of the Picadilly and Orange boats.
- Alternative hypothesis (Ha): There is a difference in the mean times of the Picadilly and Orange boats.
3. Test Statistic:
- The t-test statistic is calculated as -3.76. This value represents how many standard errors the difference between the means is away from the hypothesized difference of 0.
4. p-value:
- The p-value is calculated as 0.0017 (two-tailed). The p-value represents the probability of obtaining a t-statistic as extreme as the observed one if the null hypothesis is true. In this case, the p-value is less than the significance level of 0.05, suggesting strong evidence against the null hypothesis.
5. Confidence Interval:
- The 95% confidence interval for the difference in means is calculated as (-4.2304, -1.1796). This means that we can be 95% confident that the true difference in mean times falls within this range.
Overall, based on the t-test results, we can conclude that there is a significant difference in the mean times of the Picadilly and Orange boats. The Picadilly boat has a lower mean time compared to the Orange boat.