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    Calculating the Confidence Interval for Standard Deviation

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    In manufacturing tablets, the quantity of active ingredient that is required to lie within 5 mg of the mean is normally distributed and the required condition is considered to hold if sigma < 1.6. A sample of 35 tablets has a sample standard deviation s = 1.321
    (a) Find a 95% confidence interval for sigma.
    (b) Formulate and perform a suitable hypothesis test at a significance level of 0.05 to see whether the data provides evidence that the required condition is being met.
    (c) Find the power of your test in (b) when sigma = 1.2, 1.4 and 1.6, approximating the answers as best the percentage points in the tables permit.
    (d) In the given context how would you answers in (c) help to assess the usefulness of the given data?

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    Solution Preview

    (a) We need to find the confidence interval for the standard deviation. We know that that formula for the 95% interval is
    [(n-1)*s^2/x^2(0.025), (n-1)*s^2/x^2(0.975)).
    For the chi square distribution with df=n-1=35-1=34, from the table, x^2(0.025)=51.9659 and x^2(0.975)=19.8062.
    So a 95% confidence interval for sigma is [34*1.321^2/51.9659, 34*1.321^2/19.8062]=[1.1417, ...

    Solution Summary

    This solution gives a detailed step by step description for calculating the confidence interval for standard deviation and its power. All required formulas and calculations are shown and explained.