Statistical Interval for a Single Sample

1. Statistical Interval for a Single Sample Outlines: • Confidence interval on the mean of a normal distribution, variance known. • Confidence interval on the mean of a normal distribution, variance unknown. • Confidence interval on the variance and standard deviation of a normal distribution. • Large-sample confidence interval for a population proportion.

2. Confidence interval • Confidence interval: Bounds represent an interval of plausible values for a parameter. • Suppose that we estimate the mean viscosity of a chemical product to be , we do not know exactly that the mean likely to be between 900 and 1100? or 990 and 1010? • Because we use a sample from the population to compute the interval, we have high confident that it does contain the unknown population parameter.

3. Confidence interval • Practical example A machine fills cups with margarine, and is supposed to be adjusted so that the mean content of the cups is close to 250 grams of margarine. Of course it is not possible to fill every cup with exactly 250 grams of margarine. Hence the weight of the filling can be considered to be a random variable X. The distribution of X is assumed here to be a normal distribution with unknown expectation μ and known standard deviation σ = 2.5 grams. To check if the machine is adequately calibrated, a sample of n = 25 cups of margarine is chosen at random. • The sample shows actual weights , with mean: • if the population mean actually around 250g. The value of • If , population mean shouldn’t close to 250g.

4. Confidence interval (Case I) Confidence interval on the mean of a normal distribution, variance known. • Suppose that X1, X2, ...,Xnis a random sample from a normal distribution with unknown μ and known σ2 . • We known that • A Confidence interval estimate for μ is Prob. of selecting samples provide the range of µ that contains the true value of µ

5. Confidence interval (Case I) • In order to find lower and upper confidence limits:

6. Confidence interval (Case I) • Interpreting a CI • We cannot say: "with probability (1 − α) the parameter μ lies in the confidence interval." • We can say that: if an infinite number of random samples are collected and a 100(1-)% CI for µ is computed from each sample, 100(1-)% of these intervals will contain the true value of µ

7. Confidence interval (Case I) Ex. Ten measurements of impact energy on specimens of steel are: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3. Assume that impact energy is normally distributed with σ= 1J. We want to find a 95% CI for μ That is, based on the sample data, a range of highly plausible values for mean impact energy for steel is 63.84J-65.08J.

8. Confidence interval (Case I) • Choice of Sample Size

9. Confidence interval (Case I) Ex. Consider the previous example, we want to determine how many specimens must be tested to ensure that the 95% CI on μof steel has a length at most 1.0J. CI length <= 1.0J, E= 0.5J n = 16

10. Confidence interval (Case I) • One-Sided Confidence Bounds Ex. From previous Ex, find a lower one sided 95% CI for mean impact energy.

11. Confidence interval (Case I) • Large Sample Confidence Interval for μ • has any distribution, n>=40, variance unknown • We can approximate CI for μby replacing σ by S.

12. Confidence interval (Case I) • Ex

13. Confidence interval (Case I)

14. Confidence interval (Case II) Confidence interval on the mean of a normal distribution, variance unknown. • Suppose that X1, X2, ...,Xnis a random sample from a normal distribution with unknown μ and unknown σ2 . • n<40

15. Confidence interval (Case II)

16. Confidence interval (Case II)

17. Confidence interval (Case II) • Ex

18. Confidence interval (Case II)

19. Confidence interval (Case III) Confidence interval on the variance and standard deviation of a normal distribution.

20. Confidence interval (Case III) • Two-Sided CI • One-Sided CI

21. Confidence interval (Case III) • Ex

22. Confidence interval (Case IV) Large-sample confidence interval for a population proportion. Normal approximation for a Binomial Proportion

23. Confidence interval (Case IV)

24. Confidence interval (Case IV) • Ex

25. Confidence interval (Case IV) • Choice of sample size • we are at least 100(1-α)% confident that the error in estimating p by is less than E if the sample size is

26. Confidence interval (Case IV) • Ex.

27. Confidence interval (Case IV) • One-Sided CI

28. Homework 1. A confidence interval estimate is desired for the gain in a circuit on a semiconductor device. Assume that gain is normally distributed with sd.=20. • Find a 95% CI for µ when n =10 and • Find a 95% CI for µ when n =25and • Find a 99% CI for µ when n =10 and • Find a 99% CI for µ when n =25 and • How does the length of CIs computed above change with the changes in sample size and confidence level? 2. The sugar content of the syrup in canned peaches is normally distributed. A random sampling of n=10 cans yields a sample standard deviation of s=4.8 mg. Calculate a 95% two-sided confidence interval for  3. The 2004 presidential election exit polls from the critical state of Ohio provided the following results. There were 2020 respondents in the exit polls and 768 were college graduates. Of the college graduates, 412 votes for George Bush. • Calculate a 95% confidence interval for the proportion of college graduates in Ohio that voted for George Bush? • Calculate a 95% lower confidence bound for the proportion of college graduates in Ohio that voted for George Bush?