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 aircraft altimeters. Here's our problem statement: Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P-value method or the traditional method of testing hypotheses.
Company A uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in the errors listed below. Use a 5% level of significance to test the claim that the new production method has errors with a standard deviation greater than 32.2 feet, which was the standard deviation for the old production method. If it appears that the standard deviation is greater, does the new production method appear to be better or worse than the old method? Should the company take any action?
OK, the first part of this problem is asking us to identify the null and alternative hypotheses. Remember that the null hypothesis is by definition a statement of equality. So right off the bat, we can eliminate answer options B, C, and F. Now we have to choose between answer options A, D, and E. To do that, we need to form the alternative hypothesis. Typically, the alternative hypothesis reflects the claim that's being made. What is the claim being made here? Well, in our problem statement, we see that the claim is that the new production method has errors with a standard deviation greater than 32.2. So we want the answer option where sigma is greater than 32.2. That's going to be answer option A. I check my answer. Nice work!
Now we're asked to find the test statistic. To do that, I'm going to use StatCrunch. I'm going to select this icon right here and open my data set that's given to me in StatCrunch. Now that my data set is here in StatCrunch, I’m going to go into Stat –> Variance Stats (because this is the only option StatCrunch has for testing standard deviations and hypothesis testing) –> One Sample (because I have only one sample) –> With Data (because I have an actual data set).
I select the column where my data is located. In the hypothesis test area, I'm going to make sure these fields match the hypotheses that I selected from the previous part of the problem. But remember that because we're testing on variance, this is sigma squared, and our hypotheses have just sigma. So we need to take this value of 32.2 and square it to put here into the null hypothesis field. So I take out my calculator, 32.2 squared is 1036.84. And now I make sure that the inequality sign for my alternative hypotheses match. Now I'm all ready to hit Compute!, and here's my answer option.
Earlier when doing confidence intervals, you need to take the square root of the upper and lower limits that come out of the results window here. For hypothesis testing, that is not necessary. Just take the test statistic. It's the second to last value here in the table, and I put that in my answer field. Fantastic!
The next part of our problem asks us to find the critical values. To do that, I'm going to pull up the chi-square calculator. So I go to Stat –> Calculator –> Chi-square. I'm using the chi-square calculator because this is the distribution for hypothesis testing on standard deviations. My degrees of freedom is 1 less than my sample size. I have 12 values in my sample data set, so this is going to be 11. I want to find the critical value, so I need to clear out this default value in this field. My inequality sign needs to match my alternative hypothesis, which was “greater than.”
And then here in the probability field, I need to insert the significance level that I'm using to test, because the area for the critical region is the significance level for your hypothesis test. Here in the problem statement, we see we are instructed to use a 5% level of significance. So here in this probability field, I'm going to put 0.05.
I press Compute!, and here is my critical value, which is this value here bounding the critical region. Notice there's only one critical value because I have a one-tailed test, a right tailed test. Nice work!
Now we're asked to evaluate or resolve the hypothesis test. Our test statistic is 35.84, so here on my distribution curve a value of 35.84 would put me somewhere around here. So I'm definitely inside the critical region, which means I'm inside the rejection region. And therefore I'm going to reject the null hypothesis. So my test statistic is greater than the critical value; that's the only way to get inside the critical region for a right-tailed test. Therefore, I'm going to reject the null hypothesis.
Because I'm rejecting the null hypothesis, there is sufficient evidence to support the claim. I check my answer. Nice work!
And now, the final part of the problem asks us to translate the results of the hypothesis test into real-world terms. We've rejected the null hypothesis. That means we're supporting the claim that our standard deviation is greater than 32.2. 32.2, as we learned from the problem statement, is the standard deviation for the old production method. So if we have sufficient evidence to support the claim that our standard deviation is now greater than what it was before, that means we have more variation in our process than we did before.
So the variation appears to be greater than in the past, so the new method appears to be worse. The new production method is worse than the old one because you have more variation. More variation means there's more room for error to creep in, and you're going to be producing more defective products. So there will be more altimeters that have errors. And because your process is now worse, the company, yes, should take immediate action to reduce the variation. I check my answer. Nice work! Well done! All that good stuff!
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