Confidence Interval for a Proportion Adapted from North Carolina State University

Objective Construct a one- sample z-confidence intervals for a population proportion.

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 students that want more campus weekend activities

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 KEY POINT: –“p” is the population proportion (the overall proportion that we are trying to estimate) –“p-hat” is the sample proportion that we find ^

Example: Estimating an unknown population proportion p Question: Is Sidney Lowe’s departure good or bad for State's men's basketball team? In a random sample of 1000 students, 590 say that Lowe’s departure is good for the basketball team. “p-hat” = 590/1000 =.59 is the point estimate of the unknown population proportion p that think Lowe’s departure is good. KEY POINT: p and p-hat are always written as decimals. ^

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 ^ Another type of estimate

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

Conditions for Constructing a Confidence Interval 1) RANDOM: –1) if the data was collected in a SURVEY or an OBSERVATIONAL STUDY, then the data must be randomly collected using an SRS –OR –2) if the data was collected in an EXPERIMENT, then the subjects should be randomly assigned to treatments.

Conditions for Constructing a Confidence Interval 2) NORMAL –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-hat will be approximately a normal distribution. –n is large when n(p-hat) ≥ 10 and n(1 minus p-hat) ≥ 10). What Does This Mean?: n(p-hat)>10 means “at least 10 people or things met the condition you were measuring” and n(1 minus p- hat)>10 means “at least 10 people or things did not meet the condition you were measuring.”

Conditions for Constructing a Confidence Interval 3) INDEPENDENT –To verify that the dependency of the sample observations is negligible, we follow the 10% rule. The 10% rule states that our sample must not exceed 10% of the overall population. –Written as a formula, this is N > 10n, where N is the population size and n is the sample size.

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

95% Confidence Interval for p

.95 Confidence level Sampling distribution model for KEY POINT: 1.96 is only the value we use for a 95% confidence interval. KEY POINT: “q” is the same thing as (1-p). Ex: If p is.23, then q is.77).

Example (Gallup Polls)

What about other Confidence Levels? Confidence intervals other than 95% confidence intervals are also used

Standard Normal

98% Confidence Intervals

Four Commonly Used Confidence Levels Confidence Level Critical value (z*)

KEY POINT: To find the z* levels on your own, use the “invNorm” feature on your calculator. EX: For a 90% confidence interval, the area would be.05, because there is 5% below the interval (the other 5% is above). Therefore, the z* value is approximately

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

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

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

IMPORTANT The higher the confidence level, the wider the interval. Therefore, a 95% confidence interval will always be wider than a 90% confidence interval (assuming that they are based off the same sample data).

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

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.

Example (solution)