Math 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 24 pistons and determines that the standard deviation is 0.0045 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the alpha equals 0.10 level of significance? Question content area bottom Part 1 What are the correct hypotheses for this test? The null hypothesis is H Subscript 0: ▼ mu p sigma ▼ not equals less than greater than equals ▼ 0.0045. 0.005. The alternative hypothesis is H Subscript 1: ▼ p sigma mu ▼ greater than equals less than not equals ▼ 0.005. 0.0045.
Solution
To determine whether the standard deviation of the piston diameters has decreased, we need to test the hypotheses about the population standard deviation (). Here are the correct hypotheses:
Null Hypothesis ():
This means there is no change in the population standard deviation from the original standard deviation.
Alternative Hypothesis ():
This suggests the population standard deviation has decreased, which is what the manager wants to test.
Explanation of Choices:
- The parameter being tested is the standard deviation (), not the mean () or proportion ().
- The null hypothesis assumes equality () with the original value of 0.005.
- The alternative hypothesis tests whether the standard deviation has decreased, indicated by the less than sign ().
Would you like me to proceed with conducting the hypothesis test using the chi-square distribution?
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Standard Deviation
Chi-Square Test
Formulas
Chi-Square Test Formula
Degrees of Freedom: n - 1
Theorems
Chi-Square Distribution for Variance Testing
Suitable Grade Level
College Level (Statistics)
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