Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to use StatCrunch to perform hypothesis testing on standard deviations of piston diameters. Here's our problem statement: The piston diameter of a certain hand pump is 0.4 inch. The manager determines that the diameters are normally distributed with a mean of 0.4 inch and a standard deviation of 0.005 inch. After recalibrating the production machine, the manager randomly selects 25 pistons and determines that the standard deviation is 0.045 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the alpha equals 10% level of significance?
OK, the first part of our problem asks us to determine the correct hypotheses. For this test, we're looking at a test on standard deviation, so we want to select population standard deviation as our parameter.
The null hypothesis is always a statement of equality. And then the question becomes “What value do we put here?” Typically this is the claimed value. And the claim that we're making is do we have enough evidence to conclude that there's a decrease in the diameter? So if we're looking for a decrease in the diameter, the value that we want to put here is the standard diameter that we're measuring against — not the sample that we're looking at, but for the population. The standard belongs with the population. So that's the one that we want to select. Then of course, here the claim is that we're looking at a decrease from that value. Good job!
Now I'm asked to calculate the test statistic. This is the part where I would love to have StatCrunch do the heavy lifting for me. However, I'm not aware of any feature inside StatCrunch which will actually calculate this for us. We can do hypothesis testing on the variance, but the test statistic that we get is for the variance, and it cannot be manipulated or somehow used to get the test statistic for the standard deviation. So we're gonna have to go old school, unfortunately, and calculate this out by hand.
Our test statistic is computed by using this equation right here. So we've got 1 minus the sample size (so this is essentially our degrees of freedom) times our sample variance divided by our population variance. If we put the numbers from the problem statement here under this equation, this is what we get. There's 25 in our sample size, so that number goes in for n. And then our sample standard deviation is 0.0045, and then the population standard deviation, 0.005. If I go ahead and compute this out, I end up with 19.44, so there's my test statistic — 19.44. Fantastic!
Now we're asked to find the P-value. And here you can stay old-school and use a chi-square table, or you can actually use the calculator inside StatCrunch and calculate your answer that way. And that's the route I'm going to be taking.
So here in StatCrunch I'm going to click on Stat –> Calculators –> Chi-square. Now here's my chi-square distribution calculator. The degrees of freedom is one less than the sample size, so sample size of 25 gives us 24 degrees of freedom. Here we want “less than” because that matches the alternative that we're looking at; we want that left tail on our hypothesis test. And then here we're going to put in our test statistic 19.44, and then Hit Compute!
And then this is the area under the curve that's less than the test statistic. And that is the definition of the P-value. So the P-value is the area under the curve for your distribution that's bounded by the test statistic. In this case, we’re asked to round to three decimal places. Good job!
Now we're asked to evaluate the hypothesis test, make a conclusion. So the P-value we have, 27%, we're comparing that with a significance level of 10%. Definitely we're outside of that region of rejection, so 10% is this — or [rather] the significance level. We're outside the region of rejection. So the P-value is going to be greater than the level of significance, which puts us outside the region of rejection. So we fail to reject (or do not reject) the null hypothesis. And because we fail to reject the null hypothesis, there is not sufficient evidence. Good job!
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Frustrated with a particular MyStatLab/MyMathLab homework problem? No worries! I'm Professor Curtis, and I'm here to help.