Inference about a population proportion Chapter 20 © 2006 W.H. Freeman and Company
TOPICS IN Chapter 20 • The sample proportion • The sampling distribution of • NOT COVERED: Large sample CI for p • Accurate confidence intervals for p • NOT COVERED: Choosing the sample size • NOT COVERED: Significance tests for a proportion
Recall: two types of measurements • Quantitative (e.g., height, age, blood cholesterol level); numeric values can be summed and averaged • Categorical – characteristics that fall into one of several categories. (e.g., blood type, hair color, whether patient survives) • This chapter categorical measurements that fall into two categories (“binary data”)
Sample proportion We call a given categorical characteristic a “success”. The sample proportion of successes (“p-hat”) is: 50 people in an undergrad class, 10 of them are Hispanic: = (10)/(50) = 0.2 (proportion of Hispanics in sample) 120 Herpes patients given a new drug; 30 get better:= (30)/(120) = 0.25 (proportion improving in sample)
Sampling distribution of In large samples, the sampling distribution of is approximately Normal .
Conditions for inference on p • Data regarded as SRS from the population. • Sample size n is large How large a sample is large enough? ANS: Different procedures require sample sizes the inferential procedure we are going to cover is suitable when n ≥ 10
“Plus four” confidence interval for p Pretend you have four additional observations, two of where are successes and two are failures
Medication side effects 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 reactions (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) 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 Irregular menstrual periods
Illustrative Example In studying 440 patients, 23 experience an adverse reaction. Therefore, the sample proportion is: Note: ≥ 4 significant digits during calculations Let’s calculate a 90% confidence interval for the population proportion p of arthritis patients who suffer some “adverse symptoms.”
90% CI for population proportion p Between 3.8% and 7.4% of arthritis patients taking this pain medication will experience adverse symptoms (said with 90% confidence).