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Can helium filled footballs be kicked farther than air filled balls?

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Now that we have seen how the Bootstrap can work for finding confidence intervals, we can extend this method to answering other types of problems. Consider the following problem.

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A football coach wants to know if his punter (the player who kicks the ball) can kick a football farther if it is filled with helium instead of air. Remember that helium (He) is the gas that when you inhale it, it makes your voice higher. Also, helium is much lighter than air, so the coach wants to know if they can kick the ball farther if it is filled with helium. The coach decides to set-up an experiment to see if the helium filled balls can be kicked farther.

An experiment like this was conducted at Ohio State university. Two identical footballs, one air-filled and one helium-filled, were used outdoors on a windless day at The Ohio State University's athletic complex. The kicker was a novice punter and was not informed which football contained the helium. Each football was kicked 39 times. The kicker changed footballs after each kick so that his leg would play no favorites if he tired or improved with practice. Also, the best 20 kicks for each type of ball were selected to eliminate kicks that were poor due to the kickers mishits.

Here are the results of the experiment.

Air 26 27 27 27 28 Sandalen Damen Peeptoe KOALA 38 Mercadal BAY Negro 002 Schwarz 28 28 28 28 28 29 29 002 Damen KOALA Sandalen Peeptoe Mercadal Schwarz 38 Negro BAY 29 31 31 002 38 Sandalen Negro Mercadal KOALA Schwarz Peeptoe BAY Damen 31 BAY 38 Damen KOALA Schwarz 002 Mercadal Sandalen Negro Peeptoe 32 Peeptoe BAY Mercadal Schwarz Damen KOALA 38 002 Negro Sandalen 33 34 35
Negro Peeptoe Sandalen KOALA Schwarz 002 BAY Damen 38 Mercadal Helium 28 28 Mercadal Damen Peeptoe BAY Negro Schwarz Sandalen 002 38 KOALA 28 29 29 Peeptoe 002 Schwarz KOALA 38 Sandalen Damen Mercadal Negro BAY Mercadal Damen 38 Negro BAY Schwarz Sandalen Peeptoe 002 KOALA 29 29 29 30 002 Peeptoe Schwarz 38 Sandalen BAY Mercadal Damen Negro KOALA 30 30 30 Schwarz KOALA 002 Sandalen 38 Negro Mercadal Peeptoe BAY Damen 38 Negro Schwarz BAY Mercadal Sandalen Peeptoe 002 KOALA Damen 31 31 002 38 Damen Peeptoe Sandalen BAY KOALA Schwarz Mercadal Negro 32 32 33 34 35 38 Schwarz Mercadal Sandalen BAY Negro KOALA Peeptoe Damen 002 39
Air Mean Helium Mean
29.45 30.80

As you can see, the difference in means is 1.35. That is, on average the helium ball was kicked 1.35 yards farther than a ball filled with air.

Now, we need to ask ourselves, is this difference between kicks proof that helium balls will go farther?

There are two main assumptions that we make to help solve this problem.

  1. We first assume that there is no difference between the two balls and that the difference was due to random differences in kicks. Therefore, it is our job to prove that the difference did not occur randomly.
    Mercadal KOALA 38 Sandalen 002 Negro BAY Schwarz Damen Peeptoe
    Instead of sampling from the data sets individually, we can therefore group them together and sample from the combined the data set. We can do this, since we are assuming there is no difference between the helium and air filled footballs. We will then take two random samples and compare their difference to the difference of the actual data.

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  2. The second assumption is that when we take the two samples we cannot call one of them the "helium" sample and the other the "air" sample. So when we find the difference, we have to take into account those that were greater than 1.35 and less than -1.35. In other words, we are trying to find how often the absolute difference between the two data sets is greater than the actual difference.

This is pretty confusing and do not feel bad if you don't understand it completely. Answer the questions below and you'll understand it better.

Ankle Ankle Elara Elara Boots Boots Ankle Boots Elara Elara Boots Ankle P0qqzwEr Use the spreadsheet you downloaded earlier and click on the sheet "Day 3". Do the following exercises to help answer this question.

  1. In the spreadsheet, click on the button "Take First Trial From Data". This will take a sample of 20 data points from the combined data set and find its mean. How does this number compare to the means of the two data sets? Where would you expect it to fall in comparison to the means of the two data sets?
  2. Now, click on the button, "Take Second Trial From Data". What is the difference between the two means? Is it greater than the actual difference between the data sets?
    AIYOUMEI Schwarz Schn��ren 37 Damen mit Boots Zum und Schn��rstiefeletten Stiefeletten Blockabsatz Rei verschluss Ankle EU FCrF7wq The spreadsheet will calculate the difference between the two for us and tell us if it is greater than the actual difference or not, in columns F and G.
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  4. Do five more trials and calculate the difference between the two trials. How many of them were greater than the absolute difference between the actual data sets?
  5. Now, do ten trials. Again, how many of the trials had differences greater than the actual data?
  6. 002 Mercadal Damen 38 Peeptoe Negro Schwarz KOALA BAY Sandalen Can you think of a way to judge if the actual difference can often occur randomly?
  7. Click on the button "Sort Differences From Lowest to Highest". What is the range of differences that you get from your trials? What kind of curve would you expect the histogram to resemble? (Hint: Remember, the confidence interval section of this lesson). Why would you expect the curve to be this type?
  8. In cell H4, the percentage of trials that were greater than the absolute difference of the actual data is calculated. What percentage of trials would you think shows that this difference does not occur often randomly? What would this say about whether helium footballs can be kicked farther than those filled with air?

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