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Work Shown:
[tex]\sigma[/tex]= sigma = population standard deviation
The hypotheses are
[tex]H_0: \sigma = 1.45\\H_1: \sigma \ne 1.45[/tex]
We have a two tailed test as indicated by the alternative hypothesis. So there will be 2 chi-square critical values.
We want to find values of A and B such that
[tex]P(A < \chi^2 < B) = 0.95[/tex]
note how [tex]C = 1-\alpha = 1-0.05 = 0.95[/tex] is the confidence level.
The two tails combine to form alpha = 0.05, so half of that is alpha/2 = 0.05/2 = 0.025
Each tail has an area of 0.025
The chi-square distribution is NOT symmetric, so we cannot use tricks earlier done with the normal distribution.
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Use a chi-square table to find the area to the right of 45.722 is roughly 0.025, so [tex]P(\chi^2 > 45.722) \approx 0.025[/tex], therefore B = 45.722 approximately.
The degrees of freedom are df = n-1 = 30-1 = 29.
With the same degrees of freedom, use the table to find that [tex]P(\chi^2 < 16.047) \approx 0.025[/tex] which shows that A = 16.047 approximately.
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[tex]P(A < \chi^2 < B) = 0.95[/tex]
updates to
[tex]P(16.047 < \chi^2 < 45.722) \approx 0.95[/tex]
The two critical values are 16.047 and 45.722
Answer:
Answer: 16.047 and 45.722 (Choice D)
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