Applying one-way ANOVA hypothesis testing to bicycle lap times

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to apply one-way ANOVA hypothesis testing to bicycle lap times.  Here's our problem statement: A certain statistic constructor participates in triathlons.  The accompanying table lists times in minutes and seconds he recorded while riding a bicycle for five laps through each mile of a three mile loop.  Use a 5% significance level to test the claim that it takes the same time to ride each of the miles.  Does one of the miles appear to have a hill?

Part 1

OK, the first part of this problem asks us to determine our null and alternative hypotheses.  With one-way ANOVA testing, the null hypothesis is always going to be that all of our population parameters are the same; they're equal to each other.  The alternative hypothesis is always going to be that at least one of those population parameters is going to be different.  This is pretty much set in stone, the way that it is.  So when you got one-way ANOVA testing, you know that this is the way it's going to turn out.  Well done!

Part 2

Now the next part of this problem asks us to find the F test statistic.  We can do this by taking the data and dumping it into StatCrunch.  So here's my data.  I have it here in StatCrunch.  I'm going to resize this window so we can see a little bit better what's going on.

Notice here how they took the times for the different laps, and they basically broke it out, dividing minutes from seconds into different columns.  This is very useful for us.  There's a note here that says we need to convert everything over to either minutes or seconds.  And if you had the minutes and the seconds in the same column, that is what you would need to do because you've got two different units for time, and the test doesn't work unless all the units are consistent.  So you'd have to convert everything to one or the other.  Here where the minutes and seconds are actually separated out, we don't need to do that because notice all of the minutes are 3 minutes.  And because all the minutes are the same, that means, up to 3 minutes, all the times are the same.  They're equal, so we don't even need to consider that.  We just need to consider the portion that's not equal, which are the seconds.  So because all the minutes are 3 minutes, we can get away with just using the seconds columns for our test to conduct the test.

We're going to go to Stat --> ANOVA --> One way.  Here I'm going to select the columns that have all the seconds in them.  And to select more than one column, I hold down the Ctrl button on my keyboard while I select additional columns.  And that's all I need to do.  I hit Compute!, and here's my results window.  And down here in my ANOVA table is my F statistic, which is what I'm looking for.  I'm asked around to four decimal places.  Nice work!

Part 3

Next we're asked to find the P-value.  The P-value is here in the ANOVA table right next to our test statistic.  Well done!

Part 4

What is the conclusion for this hypothesis test?  Well, we were asked to use a 5% significance level, and 5% definitely compared with the P-value, the P-value of --- what is that? Like 1% of 1%?  That's definitely less than 5%, so we're going to be inside the region of rejection.  And therefore we reject the null hypothesis.  Every time we reject the null hypothesis, there is always sufficient evidence.  Good job!

Part 5

And now the last part of the problem asks us, does one of the miles appear to have a hill?  Well, we rejected the null hypothesis.  What does that mean?  Well, go back to the null hypothesis.  The null hypothesis says that everything is the same, that they're all equal.  We're rejecting that, which means this is not true. At least one of the three is going to be different from the others.

We come over here to our results window and look at the mean values that are listed for each of the different miles.  Notice Mile 3 looks significantly longer than Mile 1 and Mile 2.  Mile 1 and Mile 2 are not exactly the same but in the same neighborhood.  This is in a neighborhood entirely different.  So it looks like Mile 3 may, indeed, have a hill.

So let's look at our options here.  "Yes, data suggest the third mile appears to take longer and a reasonable expectation is that it has a hill."  Good job!

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