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Confidence Intervals for Unknown Proportion p

Learn how to determine confidence intervals for an unknown population proportion p based on a single sample and how to interpret them correctly. Also, explore how to calculate the required sample size for estimating p with a specified margin of error and confidence level.

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Confidence Intervals for Unknown Proportion p

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  1. From the Data at Hand to the World at LargeChapter 18Confidence Intervals for an Unknown Population p Estimation of a population parameter: Estimating an unknown population proportion p

  2. Chapter 18 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. • Determine the required sample size to estimate an unknown population proportion p to within a specified margin of error with a specified level of confidence.

  3. Warmup 1: How do social networking sites impact daily lives and relationships? • Recent Pew Research survey: of the 156 respondents age 18-22 who use Facebook, 48 said that they update their status daily. • What can we say about the population proportion p= proportion of all 18-22 yr old Facebook users who update their status daily?

  4. Warmup 2

  5. 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

  6. 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

  7. Example: Estimating an unknown population proportion p • Is Mark Gottfried’s departure good or bad for State's men's basketball team? (News & Obs. online poll; not scientifically valid!!) • Out of 1576 responses, 790 said that Gottfried’s departure was good for the bb team. • p = 790/1576 = .501 is the point estimate of the unknown population proportion p that think Gottfried’s departure is good for the bb team. ^

  8. Another type of estimate Shortcoming of Point Estimates ^ • p = 790/1576 = .501, best estimate of population proportion p BUT How good is this best estimate? No measure of reliability

  9. Interval Estimator A confidence interval is a range (or an interval) of values used to estimate the unknown value of a population parameter . Opinion Poll

  10. 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 ≥ 15 and nq ≥ 15).

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

  12. 95% Confidence Interval for p

  13. Confidence level Sampling distribution model for .95

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

  15. Warmup 1: How do social networking sites impact daily lives and relationships? 48 of 156 respondents age 18-22 who use Facebook said that they update their status daily. What can we say about the population proportion p = proportion of all 18-22 yr old Facebook users who update their status daily?

  16. Warmup 2

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

  18. Standard Normal

  19. 98% Confidence Intervals

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

  21. 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

  22. 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.

  23. 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

  24. 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)

  25. 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 347 strongly favor the current system.

  26. Example (solution)

  27. Interpreting Confidence Intervals • Previous example: .347±.0295(.3175, .3765) • Correct: We are 95% confident that the interval from .3175 to .3765 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. (.3175, .3765) 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 .3175 and .3765. (Note that p is not random, it is a fixed but unknown number)

  28. Confidence Interval Interpretation 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? 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.

  29. Determining the Sample Size n to Estimate a Population Proportion p

  30. 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.

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

  32. Confidence level Sampling distribution of .95

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

  34. 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

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

  36. 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?

  37. 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:

  38. 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

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