Does a football filled with helium travel farther than one filled with ordinary air? To test this, the Columbus Dispatch conducted a study. Two identical footballs, one filled with helium and one filled with ordinary air, were used. A casual observer was unable to detect a difference in the two footballs. A novice kicker was used to punt the footballs. A trial consisted of kicking both footballs in a random order. The kicker did not know which football (the helium-filled or the air-filled football) he was kicking. The distance of each punt was recorded. Then another trial was conducted. A total of 39 trails were run. Here are the data for the 39 trials, in yards that the footballs traveled. The difference (helium minus air) is the response variable.
Helium 25 16 25 14 23 29 25 26 22 26
Air 25 23 18 16 35 15 26 24 24 28
Difference 0 -7 -7 -2 -12 14 -1 2 -2 -2
Helium 12 28 28 31 22 29 23 26 35 24
Air 25 19 27 25 34 26 20 22 33 29
Difference -13 9 1 6 -12 3 3 4 2 -5
Helium 31 34 39 32 14 28 30 27 33 11
Air 31 27 22 29 28 29 22 31 25 20
Difference 0 7 17 3 -14 -1 8 -4 8 -9
Helium 26 32 30 29 30 29 29 30 26
Air 27 26 28 32 28 25 31 28 28
Difference -1 6 2 -3 2 4 -2 -2 -2
(a) Examine the data. Is it reasonable to use the t procedures?
(b) If your conclusion in part (a) is “Yes,� do the data give convincing evidence that the helium-filled football travels farther than the air-filled football?
a) yes. b)no.
(a) To determine if it is reasonable to use the t procedures, we need to check if the conditions for using t procedures are met.
1. Randomization: The description states that the footballs were kicked in a random order, so this condition is satisfied.
2. Independence: It is not explicitly mentioned how the footballs were chosen for each trial. Assuming that the same footballs were not used multiple times and that each trial had a randomly selected football, we can assume independence.
3. Nearly Normal: For each of the helium and air football distances, we need to verify if the distribution is approximately normal. Given that the sample size is 39, we can proceed with checking the normality assumption.
To assess the normality assumption, we can visually examine the distribution of the differences between the helium and air football distances using a histogram or a normal probability plot.
(b) To determine if there is convincing evidence that the helium-filled football travels farther than the air-filled football, we need to perform a hypothesis test.
Null Hypothesis (H0): The mean difference in distances between helium and air footballs is zero.
Alternative Hypothesis (Ha): The mean difference in distances between helium and air footballs is greater than zero.
We can use a one-sample t-test for the mean difference. We need to compute the test statistic and compare it to the critical value based on the significance level (alpha) to make a conclusion.
If the conditions for using t procedures are met and the test statistic falls in the rejection region (after comparing with the critical value), we would have convincing evidence that the helium-filled football travels farther than the air-filled football.
(a) In order to determine whether it is reasonable to use t procedures, we need to assess whether the conditions for using these procedures are met:
1. Independence: The independence condition is satisfied since each punt is considered a separate trial and the kicks are random.
2. Nearly Normal Distribution: In order to assess this condition, we can create a histogram or boxplot of the differences in yardage between the helium-filled and air-filled footballs. If the distribution is approximately symmetric and unimodal, this condition is likely met.
(b) To determine if there is convincing evidence that the helium-filled football travels farther than the air-filled football, we can perform a hypothesis test using the data provided. The null hypothesis, H0, would be that there is no difference in the distances traveled by the two footballs (i.e., the mean difference is zero), while the alternative hypothesis, Ha, would be that the helium-filled football travels farther than the air-filled football (i.e., the mean difference is greater than zero).
To perform the hypothesis test, we can use a one-sample t-test for the mean difference of paired data. We would calculate the test statistic and p-value, and compare the p-value to our chosen significance level (e.g., 0.05) to make a conclusion.
In this case, due to the limited information provided (only the differences in yardage), we cannot perform the necessary calculations to determine the test statistic and p-value. However, by collecting this additional information and performing the appropriate calculations, we can determine whether there is convincing evidence that the helium-filled football travels farther than the air-filled football.