Concept explainers
Let α1 > 0, α2 > 0, with α1 + α2 = α. Then
- a. Use this equation to derive a more general expression for a 100(1 - α)% CI for μ of which the interval (7.5) is a special case.
- b. Let α = .05 and α1 = α /4, α2 = 3α/4. Does this result in a narrower or wider interval than the interval (7.5)?
a.
Derive the general expression for
Answer to Problem 8E
The general expression for
Explanation of Solution
Given info:
For
Calculation:
The usual
Here, it is given that with probability
General expression for confidence interval is obtained as follows:
From the obtained result, there is evidence to infer that with probability
Therefore, the
b.
Find the width of the usual confidence interval about mean for
Find the width of the confidence interval obtained in part (a) for
Compare the two confidence intervals.
Answer to Problem 8E
The width of the usual confidence interval about mean for
The width of the confidence interval obtained in part (a) for
The confidence interval obtained in part (a) for
Explanation of Solution
Calculation:
Width of usual confidence interval for
Critical value:
The level of significance is
From Table A.3 of the standard normal distribution in Appendix , the critical value corresponding to 0.05 area to the right is 1.96.
Thus, the critical value is
The usual
The general formula to obtain width of the interval is,
Thus, the width of the usual confidence interval about mean for
Width of usual confidence interval obtained in part (a) for
Critical value:
For
The level of significance is
From Table A.3 of the standard normal distribution in Appendix , the critical value corresponding to 0.0125 area to the right is 2.24.
Thus, the critical value is
For
The level of significance is
From Table A.3 of the standard normal distribution in Appendix , the critical value corresponding to 0.0375 area to the right is 1.78.
Thus, the critical value is
The
The general formula to obtain width of the interval is,
Thus, the width of the usual confidence interval for population mean
Comparison:
The width of the usual confidence interval about mean for
The width of the confidence interval obtained in part (a) for
By observing the two widths it can be concluded that, confidence interval obtained in part (a) for
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Chapter 7 Solutions
DEVORE'S PROB & STATS F/ENG WEBASSIGN
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage