   Chapter 8, Problem 76SE

Chapter
Section
Textbook Problem

Chapter 7 presented a CI for the variance σ2 of a normal population distribution. The key result there was that the rv χ2 = (n − 1)S2/σ2 has a chi-squared distribution with n − 1 df. Consider the null hypothesis H 0 : σ 2 = σ 0 2 (equivalently, σ = σ0). Then when H0 is true, the test statistic χ2 = (n − 1)S2/ σ 0 2 has a chi-squared distribution with n − 1 df. If the relevant alternative is H a : σ 2 > σ 0 2 the P-value is the area under the χ2 curve with n − 1 df to the right of the calculated χ2 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 0.58°C.

Calculation:

Step 1: Parameter of interest:

Let σ be the true standard deviation.

Step 2: Null hypothesis:

H0:σ=0.50

That is, the standard deviation of the softening point is 0.5°C.

Step 3: Alternative hypothesis:

Ha:σ>0.5

That is, the standard deviation of the softening point is more than 0.5°C.

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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