SOLUTION: 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 h
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Question 1196735: 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 errors listed below. Use a 0.05 level of significance to test the claim that the new production method has errors with a standard deviation greater than 32.2 ft, 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?
-40
76
-21
-70
-40
11
15
52
-9
-51
-108
-108
Find the test statistic.
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Use technology or a graphing calculator to compute the sample standard deviation of the given data set.
You should get roughly s = 57.384601 as the sample standard deviation.
Be sure not to mix up population standard deviation with sample standard deviation. They're slightly different ideas.
There are n = 12 items in the data set. This is the sample size.
sigma = population standard deviation
Let's set up the two hypotheses
Null: sigma ≤ 32.2
Alternative: sigma > 32.2
The claim is in the alternative hypothesis.
Now compute the chi-square test statistic
ChiSquare = (n-1)*(s/sigma)^2
ChiSquare = (12-1)*(57.384601/32.2)^2
ChiSquare = 34.935879
What this tells us is that P(ChiSquare > 19.675) = 0.05 approximately when df = 11.
This value 19.675 is the ChiSquare critical value. Anything larger than this is in the rejection region. This fits the description for the test statistic we got earlier (34.935879)
Therefore, we reject the null and conclude that the alternative hypothesis is true. The population standard deviation sigma appears to be larger than 32.2 feet. This means that the errors are more spread out and less consistent. It will likely result in aviation accidents of some kind if the company doesn't address the issue.
Answer:
The new method appears to be worse than the old method.
The company should take action.
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