Conducting mean hypothesis testing on IQ and lead level data

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to conduct mean hypothesis testing on IQ and lead level data.  Here's our problem statement: Listed in the data table are IQ scores for a random sample of subjects with medium lead levels in their blood.  Also listed are statistics from a study done of IQ scores for a random sample of subjects with high lead levels.  Assume that the two samples are independent simple random samples selected from normally distributed populations.  Do not assume that the population standard deviations are equal.  Complete Parts A and B below.  Use a 10% significance level for both parts.

Part A

OK, Part A asks us to test the claim that the mean IQ scores for subjects with medium lead levels is higher than the mean for subjects with high lead levels.  And the first thing we're asked to do is to provide the null and alternative hypotheses.  We're also instructed to assume that Population 1 consists of subjects with medium lead levels and Population 2 consists of subjects with high lead levels.

So the null hypothesis is going to be a statement of equality, which it always is by definition.  So we're going to be looking at Answer options B and D.  And then to select the one with the proper alternative hypothesis, we look at this assumption that we were given here at the very end of our problem statement.  Population 1 is — has the subjects with the medium lead levels; Population 2 are the subjects with the high lead levels.  And we're testing the claim that the medium lead levels have a higher mean score than the high level.  So the medium lead levels, [which] are Population 1, will have a higher mean score than the high lead levels, [which] are Population 2.  So 1 will be greater than 2, and that's what we see here with Answer option B.  The mean of Population 1 is greater than the mean of Population 2, so that's what we'll select for our answer.  Good job!

Now the next part of Part A asks us to provide the test statistic, and we can do this inside StatCrunch.  Notice when we look at our data here, they're going to make us work a little bit here.  We've got summary statistics for the high lead level population, or — excuse me, the high lead levels sample.  But here for the medium lead level sample, we've got actual data. So we can get around this pretty easy.  All we have to do is just calculate summary statistics for the medium lead level group, and then we've got summary stats for both groups.  And we can use those summary stats to conduct our hypothesis test.

So the first step I'm going to do is put this data for that first sample here in StatCrunch.  I'm going to resize this window so we can see better everything that's going on.  Now here in StatCrunch, I'm going to get my summary stats by going to Stat --> Summary Stats --> Columns.  I select that column of data.  I want to calculate the sample size, sample mean, and the sample standard deviation.  And here's my numbers right here.  So I'm going to move this results window down to the bottom.

Now we have everything we need to calculate the test statistic by performing our hypothesis test.  So to do that, I'm going to go to Stat, I go to T Stats (because we don't know what the population standard deviation is), Two Samples (because we have two independent samples), and we have With Summary (because we don't have actual data for both samples, but we do have summary statistics for both of the samples).

Here in Sample 1 that — remember that was defined as the medium lead level group, and those are the summary stats that we just calculated a moment ago.  So I'm going to put those numbers here.  Actually the sample mean is 90.81.  And notice I'm rounding to three decimal places because here those equivalent values that were given here and these summaries are rounded to three decimal places.  I do the same thing with the standard deviation.

And now I put in the sample statistics for the second sample.  Now I scroll down here, make sure this button for Hypothesis test is selected --- that is the default, and that is what we want.  This area here needs to match what we got over here with our non alternative hypothesis, so I need to change this inequality sign. And now I'm ready. I hit Compute!, and here's my test statistic, second to last number there in my results window.  I'm asked around to two decimal places.  Good job!

Now the next part asks for the --- the next part is asking for the P-value.  That's the last value there in the data table right next door to the test statistic.  I'm asked to round to three decimal places.  Excellent!

OK, now I'm asked to state the conclusion from my test.  A P-value of over 30%, we're using a significance level of 10% --- 30% is over 10%, so we're outside the reason of rejection, which means we fail to reject the null hypothesis.  And every time we fail to reject the null hypothesis, there is not sufficient evidence.  So we don't want Answer option D because that says there is sufficient evidence.  We want Answer option A because we failed to reject the null hypothesis, and whenever we do that, there's not sufficient evidence.  Good job!

Part B

Now Part B of this problem asks us to construct a confidence interval, which we can do reasonably well enough.  Go back to your options window here, and I'm going to scroll down and switch this radio button down to confidence interval.  And I need to put in a confidence level.  Normally we would take our significance level and subtract it from 100%, but here we've got two alpha that we have to subtract because we've got two samples, so we're looking at, yeah, 20% that we subtract from 100%.  So that gives us a confidence level of 80%.  And there's my lower and upper limits right there in the results window.  So all I have to do is just transfer those numbers over.  Well done!

Part C

And now for the final question: Does the confidence interval support the conclusion of the test?  Well, let's take a look at where's zero in our confidence interval.  Is it inside or outside?  Zero is inside our confidence interval, and so therefore it's possible that these two values could be the same.  It's possible that the difference here could be zero.  And if the difference is zero, that means these two could be the same.  Well, if they're the same, then that's exactly what we set up here in our null hypothesis.  So if they're the same, then that means the null hypothesis could be true.  And if the null hypothesis is true, we don't want to reject it.  So we're going to fail to reject the null hypothesis.  And that's exactly the conclusion that we made right here.  So yes, the confidence interval does support the conclusion of the test because zero is found inside the confidence interval.  Fantastic!

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