Chapter 7 Confidence Intervals (application of the Central Limit Theorem)

1. Chapter 7Confidence Intervals(application of the Central Limit Theorem) Introduction and Confidence Intervals for Proportions

2. Chapter 7Objectives Construct confidence intervals for population proportions based on the information contained in a single sample.

3. Concepts of Estimation • The objective of estimation is to estimate the unknown value of a population parameter, like a population proportion p, on the basis of a sample statistic calculated from sample data.  e.g., NCSU student affairs office may want to estimate the proportion of studentsthat want more campus weekend activities • There are two types of estimates • Point Estimate • Interval estimate

4. Point Estimate of p ^ • p = , the sample proportion of x successes in a sample of size n, is the best point estimate of the unknown value of the population proportion p

5. Example: Estimating an unknown population proportion p • Is Sidney Lowe’s departure good or bad for State's men's basketball team? (Technician poll; not scientifically valid!!) • In a sample of 1000 students, 590 say that Lowe’s departure is good for the bb team. • p = 590/1000 = .59 is the point estimate of the unknown population proportion p that think Lowe’s departure is good. ^

6. Another type of estimate Shortcoming of Point Estimates ^ • p = 590/1000 = .59, best estimate of population proportion p BUT How good is this best estimate? No measure of reliability

7. Interval Estimator • A confidence interval is a range (or an interval) of values used to estimate the unknown value of a population parameter . • http://abcnews.go.com/US/PollVault/

8. Tool for Constructing Confidence Intervals: The Central Limit Theorem • If a random sample of n observations is selected from a population (any population), and x “successes” are observed, then when n is sufficiently large, the sampling distribution of the sample proportion p will be approximately a normal distribution. • (n is large when np ≥ 10 and nq ≥ 10).

9. Standard Normal P(-1.96  z  1.96) =. 95

10. 95% Confidence Interval for p

11. Confidence level Sampling distribution model for .95

12. Example (Gallup Polls) http://abcnews.go.com/US/PollVault/story?id=145373&page=1

13. Confidence intervals other than 95% confidence intervals are also used

14. Standard Normal

15. 98% Confidence Intervals

16. Four Commonly Used Confidence Levels Confidence LevelMultiplier .90 1.645 .95 1.96 .98 2.33 .99 2.576

17. Medication side effects (confidence interval for p) Arthritis is a painful, chronic inflammation of the joints. An experiment on the side effects of pain relievers examined arthritis patients to find the proportion of patients who suffer side effects. What are some side effects of ibuprofen? Serious side effects (seek medical attention immediately): Allergic reaction (difficulty breathing, swelling, or hives), Muscle cramps, numbness, or tingling, Ulcers (open sores) in the mouth, Rapid weight gain (fluid retention), Seizures, Black, bloody, or tarry stools, Blood in your urine or vomit, Decreased hearing or ringing in the ears, Jaundice (yellowing of the skin or eyes), or Abdominal cramping, indigestion, or heartburn, Less serious side effects (discuss with your doctor): Dizziness or headache, Nausea, gaseousness, diarrhea, or constipation, Depression, Fatigue or weakness, Dry mouth, or Irregular menstrual periods

18. 440 subjects with chronic arthritis were given ibuprofen for pain relief; 23 subjects suffered from adverse side effects. Calculate a 90% confidence interval for the population proportion p of arthritis patients who suffer some “adverse symptoms.” What is the sample proportion? For a 90% confidence level, z* = 1.645.  We are 90% confident that the interval (.034, .070) contains the true proportion of arthritis patients that experience some adverse symptoms when taking ibuprofen.

19. Example: impact of sample size n=440: width of 90% CI: 2*.018 = .036 n=1000: width of 90% CI: 2*.007=.014 When the sample size is increased, the 90% CI is narrower

20. IMPORTANT • The higher the confidence level, the wider the interval • Increasing the sample size n will make a confidence interval with the same confidence level narrower (i.e., more precise)

21. Example • Find a 95% confidence interval for p, the proportion of NCSU students that strongly favor the current lottery system for awarding tickets to football and men’s basketball games, if a random sample of 1000 students found that 50 strongly favor the current system.

22. Example (solution)

23. Interpreting Confidence Intervals • Previous example: .05±.014(.036, .064) • Correct: We are 95% confident that the interval from .036 to .064 actually does contain the true value of p. This means that if we were to select many different samples of size 1000 and construct a 95% CI from each sample, 95% of the resulting intervals would contain the value of the population proportion p. (.036, .064) is one such interval. (Note that 95% refers to the procedure we used to construct the interval; it does not refer to the population proportion p) • Wrong: There is a 95% chance that the population proportion p falls between .036 and .064. (Note that p is not random, it is a fixed but unknown number)

24. Determining Sample Size to Estimate p

25. To Estimate a Population Proportion p • If you desire a C% confidence interval for a population proportion p with an accuracy specified by you, how large does the sample size need to be? • We will denote the accuracy by ME, which stands for Margin ofError.

26. Required Sample Size n to Estimate a Population Proportion p

27. Confidence level Sampling distribution of .95

28. What About p and q=1-p?

29. Example: Sample Size to Estimate a Population Proportion p • The U. S. Crime Commission wants to estimate p = the proportion of crimes in which firearms are used to within .02 with 90% confidence. Data from previous years shows that p is about .6

30. Example: Sample Size to Estimate a Population Proportion p (cont.)

31. Example: Sample Size to Estimate a Population Proportion p The Curdle Dairy Co. wants to estimate the proportion p of customers that will purchase its new broccoli-flavored ice cream. Curdle wants to be 90% confident that they have estimated p to within .03. How many customers should they sample?

32. Example: Sample Size to Estimate a Population Proportion p (cont.) • The desired Margin of Error is ME = .03 • Curdle wants to be 90% confident, so z*=1.645; the required sample size is • Since the sample has not yet been taken, the sample proportion p is still unknown. • We proceed using either one of the following two methods:

33. Example: Sample Size to Estimate a Population Proportion p (cont.) • Method 1: • There is no knowledge about the value of p • Let p = .5. This results in the largest possible n needed for a 90% confidence interval of the form • If the proportion does not equal .5, the actual ME will be narrower than .03 with the n obtained by the formula below. • Method 2: • There is some idea about the value of p (say p ~ .2) • Use the value of p to calculate the sample size