Using bootstrap methods to construct a proportion confidence interval estimate

Intro

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use bootstrap methods to construct a proportion confidence interval estimate.  Here's our problem statement: In a study of the accuracy of fast food drive through orders, a restaurant had 38 orders that were not accurate among 451 orders observed.  Use the bootstrap method to construct a 90% confidence interval estimate of the proportion of orders that are not accurate.  Use the 200 accompanying bootstrap samples.  How does the result compare to the 90% confidence interval 0.063 to 0.106 constructed using the sample proportion?

Part 1

OK, the first part of this problem asks for a confidence interval estimate. And to construct that, we're going to take the bootstrap samples that they give us and use percentiles to construct the confidence interval.  To do that, we need to take our data.  And normally we put it into StatCrunch, but since we're doing percentiles I find it easier to do that in Excel. I'm going to put the data in Excel, which I've already done here.  And now we can see here in Excel, here's our data.

So for finding percentiles, the first thing we need to do is sort our data.  Here in Excel, we can do that by going up to Data and then in the Data ribbon we select Sort.  And now we want to sort by that second column "Not Accurate."  And we want to order it from smallest to largest.  So I press OK.  Now my data has been reordered.

We need to take these numbers here, which are counts of orders that are not accurate, and we need to convert them into proportions because we're building a confidence interval estimate on proportions, not counts.  So to transform each of these numbers, we're going to let Excel do the calculation for us.  And to do that, I'm going to put my --- I'm going to select the cell right next to that first sample row.  And I'm going to type in the formula that I want Excel to calculate.  So I start out by pressing equals, and then I'm going to use my left arrow key to select that cell just to the right, and then we're going to divide by the total number of observations from the orders (which we see here in the problem statement is 451).  I press Enter, and there Excel has calculated everything for me.

Now notice here it says we want to round our values to three decimal places.  And I can do that in Excel automatically.  I just come here, and I'm going to go back to Home, and then here in my Number area, I'm going to use this arrow right here to decrease the decimal to three decimal places.  And lo and behold, Excel has got everything rounded for me, and this is great!  All I gotta do is just read the numbers when we're all said and done.

OK, now to copy this formula down, I can do it in either one of two ways.  I can either drag it down --- notice when I take the cursor over that right bottom corner of my cell, it changes into this smaller cross.  When it's at that smaller cross, I can hold down the left button on my mouse, and then I can drag the formula down, and it'll copy everything for me.  Now there's 200 of these that I need, so I'm going to be dragging for awhile.  It's not the most efficient way to do this.

The more efficient way to do this would be to select this column here --- I mean, excuse me, this cell --- and then I'm going to hit Ctrl + C to copy it.  And then I'm going to go over one to the left.  And then to get down to the very bottom, I'm going to press Ctrl + Down Arrow, and notice it instantly brought me down to the bottom of that row.  I'm going to go back one because I want my values there in Column C.  And now I'm going to press Ctrl + Shift + Up Arrow.  So now I've not only moved my cursor up to the top of the column there, but now I've also selected everything in between, and that's where I want these values to appear.  Now I have to just paste it in.  And I do that by pressing Ctrl + V.  E viola!  There's everything copied in.

So now we've got everything that we need to calculate our percentiles.  But in order for our count to not be off so much, we're going to delete this first row that we see here.  So the first thing I'm going to do is select it by clicking on the left mouse here.  And then while my cursor is still in that same position, I'm going to right click on my mouse, and I hit Delete.  And now that first row has been deleted.

So now we can use percentiles to get our confidence interval estimate.  Our confidence level is 90%.  That means 90% of these values are going to be inside our confidence interval, which means the remaining 10% is out.  Half of that 10% is going to be on the left, and half is going to be on the right.  So 5% is going to be here on the left outside of the confidence interval.  And what does that translate to as far as how many rows do we need to count down?  Well, 5% of the 200 rows that we have --- if I get my calculator out, I got 5% of the 200 rows gives me 10.  So I need to go down to Row 10.  Here's Row 10.  And because we're using percentiles, we want the number in the next row.  So here in Row 11, the number I need his 0.06.  That's my lower limit for my bootstrap confidence interval.

To get the upper limit, we're going to do the same thing but going from the bottom up.  So here we went from --- here we went to 10.  And now we're going to go back into Excel, and I'm going to go down to the bottom of my list.  And I want to go to Row 190 --- 90 is 10 less than 200.  And the number I need is in the next row down, which is 0.111.  Nice work!

Part 2

And now the second portion of this problem asks, "If some portion of the confidence intervals overlap, the confidence intervals are not significantly different. The results of the two intervals are" --- then there's a blank.  So if you click on the drop down, then you we can see that they either are or are not significantly different.  Well, that's defined here as whether or not they overlap.  So if we look here up here at the top to see if they overlap or not, look at their intervals here.  Notice what we have here.  We've got here in this interval, we've got 0.063 is our lower limit.  Here we got 0.06.  And then the upper limit here we've got 0.106, and here we've got 0.111.  So there's quite a bit of overlap between these two confidence intervals.  And that means they're not significantly different.

Now the second part here says the interval from the bootstrap method is --- and we have to look at our interval and compare it with the actual interval that we have there.  So here we've got 0.063, 0.060.  So this is going to be on the left side of this interval.  And then 0.111 is going to be to the right of our upper limit here (0.106).  So this confidence interval that we constructed with bootstrapping is wider than the other that we were given there on the problem statement.  I check my answer.  Good job!

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