Estimation of a population parameter: Estimating an unknown population proportion p

1. From the Data at Hand to the World at LargeChapter 19Confidence Intervals for an Unknown Population p Estimation of a population parameter: Estimating an unknown population proportion p

2. Chapter 19 Objectives • Determine, manually and using technology, confidence intervals for an unknown population proportion p based on the information contained in a single sample. • Understand and properly interpret a confidence interval for a population proportion p.

3. What do we frequently need to estimate? • An unknown population proportion p • An unknown population mean  • Will deal first with estimating a population proportion p ? p?

4. 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

5. 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

6. 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. ^

7. Another type of estimate Shortcoming of Point Estimates • = 590/1000 = .59, best estimate of unknown population proportion p of students that think Lowe’s departure is good for the team. BUT How good is this best estimate? No measure of reliability

8. 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/

9. 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).

10. 95% Confidence Interval for p

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

12. Confidence level Sampling distribution model for .95

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

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

15. Standard Normal

16. 98% Confidence Intervals

17. Four Commonly Used Confidence Levels Confidence LevelMultiplier .90 1.645 .95 1.96 .98 2.33 .99 2.58

18. 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

19. 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.

20. 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

21. 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)

22. Example • Find a 95% confidence interval for p, the proportion of small businesses in favor of a tax increase to decrease the national debt, if a random sample of 1000 found the number of businesses in favor of increased taxes was 50.

23. Example (solution)

24. 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)

25. Confidence Interval Interpretation When my daughter was learning multiplication with flash cards … 7 x 5 = 42 … she achieved 95% accuracy, so she would answer 95 out of 100 correctly, say. Is there a 95% chance that 7x5=42? Of course not! 95% confidence intervals behave the same way. An individual confidence interval either captures p or it doesn’t… …but in a group of many 95% confidence intervals, about 95% of them will capture p.