Chapter 8, Problem 76SE

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9th Edition

Jay L. Devore

ISBN: 9781305251809

Textbook Problem

Chapter 7 presented a CI for the variance *σ*^{2} of a normal population distribution. The key result there was that the rv *χ*^{2} = (n − 1)*S*^{2}/*σ*^{2} has a chi-squared distribution with *n* − 1 df. Consider the null hypothesis
*σ = σ*_{0}). Then when *H*_{0} is true, the test statistic *χ*^{2} = (*n* − 1)*S*^{2}/
*n* − 1 df. If the relevant alternative is
*P*-value is the area under the *χ*^{2} curve with *n* − 1 df to the right of the calculated *χ*^{2} value. To ensure reasonably uniform characteristics for a particular application, it is desired that the true standard deviation of the softening point of a certain type of petroleum pitch be at most .50°C. The softening points of ten different specimens were determined, yielding a sample standard deviation of .58°C. Does this strongly contradict the uniformity specification? Test the appropriate hypotheses using *α* = .01. [*Hint*: Consult Table A.11.]

To determine

Test whether the given standard deviation strongly contradict the uniformity specification or not.

Explanation

**Given info:**

The softening points of ten different specimens were determined. The sample standard deviation is

**Calculation:**

*Step 1: Parameter of interest:*

Let

*Step 2: Null hypothesis:*

That is, the standard deviation of the softening point is

*Step 3: Alternative hypothesis:*

That is, the standard deviation of the softening point is more than

*Step 4: Test statistic:*

**Software procedure:**

Step by step procedure to obtain the test-statistic value using the MINITAB software:

- Choose
**Stat > Basic Statistics > 1 Variance**. - Under
**Data**, choose**Sample standard deviation**. - Enter
**Sample size**as 10 . - Enter
**Sample standard deviation**as 0.58. - Check
**Perform hypothesis test**. - Choose
**Hypothesized standard deviation**. - Enter the
**Value**as 0.5. - Select
**Options** - Enter the
**Confidence level**as**0.99**and**Alternative**as**greater than**

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