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 (σ\sigma). Here are the correct hypotheses:

Null Hypothesis (H0H_0):

H0:σ=0.005H_0: \sigma = 0.005 This means there is no change in the population standard deviation from the original standard deviation.

Alternative Hypothesis (H1H_1):

H1:σ<0.005H_1: \sigma < 0.005 This suggests the population standard deviation has decreased, which is what the manager wants to test.

Explanation of Choices:

  1. The parameter being tested is the standard deviation (σ\sigma), not the mean (μ\mu) or proportion (pp).
  2. The null hypothesis assumes equality (==) with the original value of 0.005.
  3. 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)