# Calculating the Confidence Interval for Standard Deviation

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.

Calculating Standard Deviations and Confidence Intervals.

The following table (see problem #7 on page 310 of attachment) lists the results of one question taken from a survey conducted for Joe's Bar and Grill. The Grill has recently undergone renovations, and with the new look, management has decided to change the menu. They are interested in knowing how well customers like the new menus.

(a) Calculate the mean, standard deviation, and confidence interval for menu preference assuming simple random sampling was used.

(b) Calculate the mean, standard deviation, and confidence interval, for menu preference assuming stratified sampling based on the meal eaten and used (i.e. the strata being breakfast, lunch and dinner customers). Assume that N/N = 1/2 and each stratum. How do the results compare with those you obtained with the previous sampling plan? Why?

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