1. T Confidence Intervals and Tests Weights of Football Players

2. Problem: A sample of 20 players from the Tarpon Springs High School football team is randomly selected and weighed. The data, in pounds, are given in the table below. Find a 95% confidence interval for the population mean.

3. Step 1: t Confidence interval for means Step 2: Assumptions: We have a simple random sample of football players, given in the problem. We don’t know that the population is normal so we examine the data.

4. Making a histogram of the players’ weights, there appear to be outliers. Since this would be an assumption violation we should check to see. The 5 number summary is (147, 161.5, 177, 183.5, 252). To check for outliers we find the IQR which is 183.5- 161.5 = 22. Now Q 3 +1.5(IQR) = 183.5+1.5(22)=216.5. This means that the weights 245 and 252 are outliers, so we cannot meet the assumptions for the t interval. As an exercise we will proceed.

5. Step 3: or (167.2, 194.1) 180.65 ± 13.48 Step 4: We have an assumption violation, so we are uncertain of the significance of this calculation. Aside from that, the interval from 167 to 194 tells us that we can be 95 % confident that this interval includes the true mean weight of the football player population.

6. Another way we can explain this is that this method captures the true mean 95% of the time. For each sample there is a new confidence interval and a 95% of the time the true mean is contained in the interval. Now we consider this same data for use in a test of significance.

7. Problem: It is published that the average weight of the high school football players at Tarpon Springs High School is 190 pounds. Based on the data presented earlier and repeated here, is the mean weight 190 pounds? Step 1: H 0 :  = 190 The mean weight of football players at Tarpon Springs High School is 190 pounds. H a :  ≠ 190 The mean weight of football players at Tarpon Springs High School is not 190 pounds.

8. Step 2: Assumptions--these were dealt with when we made the confidence interval and we have the same problem with outliers. We may appeal to the robustness of the t test to argue that we can use this test even with this violation. We are less certain, however, of our results when we have a violation. Step 3:

9. Step 4:Step 5: P-value = P(t 1.451) = 2(0.08154) =.1630 To make this graph on the TI-83, use the command: Shade_t(-100,-1.451,190:Shade_t(1.451,100,19)

10. Step 6: Fail to reject H 0, as a value this extreme will occur by chance alone 16% of the time. Step 7: We lack strong evidence that the mean weight of Tarpon Springs High School football players is different from 190 pounds. (We also have an assumption violation.)

11. THE END