Finding the best point estimate and confidence interval for proportions

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find the best point estimate and confidence interval for proportions.  Here's our problem statement: Use the sample data and confidence level given below to complete Parts A through D.  In a study of cell phone use and brain hemispheric dominance, an Internet survey was emailed to 2,606 subjects randomly selected from an online group involved with ears.  1064 surveys were returned.  Construct a 90% confidence interval for the proportion of returned surveys.

Part A

OK, here we've got a link to our z-score table, but that's the old school way of doing it, and it's a little calculation involved.  So we're going to be lazy (because I'm just feeling lazy today), and we're going to use StatCrunch.  So I'm going to go in and open up StatCrunch.  And it's not so much that I need any data in StatCrunch, but I just need the functionality.

So ultimately we're going to be asked to construct a confidence interval, so let's just go ahead and construct that to begin with.  I want a 90% confidence interval for the proportion of returned surveys.  So to get my confidence interval, I'm going to go to Stat --> Proportion stats (because we're dealing with proportions) --> One Sample (because I've got just the one sample) --> With Summary (because I don't have any actual data, just summary stats).

The number of successes is defined by how many successes we have where success is determined by the confidence interval here.  We want a conference interval on the proportion of returned surveys.  That means returning a survey is the definition of success.  1064 surveys were returned, so that's the number I'm going to put here in this field.  The number of observations is the total number.  That's 2606.  I want a confidence interval and a 90% confidence limit.

So here's my confidence interval.  Notice the sample proportion is calculated here for you.  You get the same number if you just go ahead and just take these numbers that you put in the options window just divide that out in your calculator.  Just take the number of successes divided by total number of observations and, see, I'm going to get the same number here.  But I just like doing it in StatCrunch because I gotta go through those steps anyway to get the confidence interval, which it's going to ask me later on the problem.  So it's just easier if I just use this.  So here's the number I need.  Good job!

Part B

Part B asks us for the margin of error.  Now some of you may be tempted to take this next number here and just use this, but that's not the margin of error.  That's a value called the standard error, which is calculated a little differently.  So this number that you see here is going to be incorrect.  How then do we get the margin of error?  Well, you can take the upper and lower limits of your confidence interval and you can average them out, but that just gives you the point estimate that's in the middle.  We already have the point estimate here.

And so if I just subtract the lower limit from the point estimate, that distance will be the margin of error.  Alternatively, I can take the upper limit and subtract out the point estimate.  That'll give me the same number.  We already have the point estimate here in the calculator, so I'm just going to go ahead and subtract out that lower limit, and also putting in all my numbers so that I can avoid rounding errors as much as possible.  And there's my margin of error.  Whoops, it rounds up.  Nice work!

Part C

And now the confidence interval just comes from the lower and upper limits there at the end of my table. So I'm going to put those in here. Good job!

Now before we move on to part D, I just want to mention that some students --- they follow the steps and they get different numbers here.  They get, you know, the software from Pearson keeps coming back to them and saying, "Hey, you got the wrong answer here."  And typically that's a result of using the wrong confidence level.  So if you're getting the wrong answers here, even though you're going through these steps, go back to your options window and make sure you've got the right confidence level in.  And more than half the time, that's the problem the students are encountering when they get the wrong answer there.

Part D

OK, Part D says, "Write a statement that correctly interprets a confidence interval. Choose the correct answer below."  A confidence interval is simply a range of values, and what we're trying to do is figure out what is the true value of the population parameter.  So when we make a confidence interval, we're saying that we have a certain level of confidence that the true value of the population parameter is between the upper and lower limits.  Here our confidence level is 90%, so we're 90% confident that the true value of the population proportion is between these two values here.  So let's look and see which answer option most, you know, best fits that statement.

Answer option A says, "One is 90% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion."  That sounds a lot like what we just went over, so let's mark that.  But before we hit Check Answer, let's go ahead and check the other answer options just to make sure.

"There's a 90% chance that the true value of the population proportion will fall between the lower bound and the upper bound."  Close, but it's not quite the same sentiment.  "90% chance of falling in." Yeah, I mean I --- it sounds like the same as 90% confidence, but you know, the way statisticians think, it's not the same thing.  So we don't want to pick that answer.

"One has 90% confidence that the sample proportion is equal to the population proportion."  No, no, that's not it either.  I mean, yes, proportions are unbiased estimators, and so the sample statistic will tend to target the population parameter.  But yeah, that's ... that's not really the same thing as saying we're 90% confident that it's lying between an upper and a lower bound.

Our final answer option says, "90% of the sample proportions will fall between the lower bound and the upper bound."  Oh yeah --- well, sort of. I mean, if you were to take an infinite number of samples and then look at the proportion and do a distribution on the proportions for those samples, then yeah, 90% of them will fall between the upper and the lower bounds.

But that's not what the confidence interval is suggesting.  The confidence interval is not suggesting infinite anything.  It's just saying we're 90% confident that the true value was between the bounds of the confidence interval.  So we're not going to select any of these other answer options.  We want this first answer option here.  Well done!

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