BIOSTATS 3.3 - CH. 19: INFERENCE ABOUT A POPULATION PROPORTION

• Chapter focuses on inference about a population proportion.
• The material is based on copyright 2018 W Freeman and Company
• The content includes previous learning objectives, learning objectives, conditions of inference, sample proportions, sampling distribution, confidence intervals for p, choosing sample size and hypothesis tests

Previous Learning Objectives

• Covers comparing two means
• Includes two-sample situations
• Two sample t procedures are relevant
• Robustness is assumed
• Pooled procedures should be avoided
• Avoid inferences on standard deviations

Learning Objectives

• How to apply inference for a population proportion
• Use the sample proportion p^
• Calculate large sample confidence intervals for a proportion
• How to find more accurate confidence intervals for a proportion
• Selection of the sample size
• Use of hypothesis tests for a proportion

Conditions for Inference on Proportions

• The data used for the estimate must come from a random sample of the population
• The population must be at least 20 times larger than the sample to ensure independence in random sampling
• The sample size n must be large enough to assume a Normal shape of the sampling distribution
• The required minimum size of n depends on the type of inference conducted

The Sample Proportion p^

• Focuses on categorical data to infer the proportion/percentage of a population with a specific trait
• If a categorical trait is labelled as a "success", the sample proportion of successes is p^
• p^ is calculated as: (count of successes in the sample) / (count of observations in the sample)
• In an example, a group of 120 Herpes patients are treated with a new drug, and 30 improve, so p^ = (30/120)
• Therefore p^ = 0.25, meaning proportion of patients improving is at 0.25 (in the sample)

Sampling Distribution of p^

• The sampling distribution of p^ is never exactly Normal
• However, it can be approximated by a Normal curve if the sample is large enough
• The mean and standard deviation (width) of the sampling distribution are determined by p and n
• Population parameter to estimate is p
• N (p,√p (1-p)/n)

Confidence Interval for p

• If p is unknown, the sampling distribution's center and spread are unknown
• A value for p has to be "guessed"
• Two options exist:
• Use p^, the sample proportion, called the large sample method, performing is poorly unless n is extremely large
• Use p̃, an improved estimate of p, called the plus four method, being reasonably accurate even for samples as small as 10

Large Sample Confidence Interval for p

• Confidence intervals contain the population proportion 'p' in C% of samples
• For a SRS (simple random sample) of size n, and with sample proportion p^calculated from the data, an approximate level C confidence interval for p is given by
• CI: p^± m, with m = zŜE = z√p^(1 – p^)/n
• Use this method when the number of successes and the number of failures are both at least 15
• Medication side effects example
• Arthritis is a painful inflammation of the joints
• Experiments tested the side effects of pain relievers with arthritis patients to determine what proportion of patients suffer side effects
• A 90% confidence interval computed calculating the population proportion of arthritis patients who suffer from "adverse symptoms”.
• Serious side effects of ibuprofen:
• Allergic reactions
• Muscle cramps, numbness, or tingling
• Ulcers in the mouth
• Rapid weight gain (fluid retention)
• Seizures
• Black, bloody, or tarry stools
• Blood in urine or vomit
• Decreased hearing or ringing in the ears
• Jaundice
• Abdominal cramping, indigestion, or heartburn
• Less serious side effects of ibuprofen
• Dizziness or headache
• Nausea, gaseousness, diarrhea, or constipation
• Depression
• Fatigue or weakness
• Dry mouth
• Irregular menstrual periods
• From a sample (n=440), 23 patients reported side effects
• p^= 23/440 = 0.052
• For a 90% confidence level, z*=1.645
• m = 1.645*√0.052 (1 – 0.052)/440 = 0.017
• 90% CI for p: 0.052 ± 0.017
• Thus, it can be determined that with 90% confidence, that between 3.5% and 6.9% of arthritis patients taking the mentioned medicine will experience some adverse symptoms

Plus Four Confidence Interval for p

• The plus four method gives more accurate confidence intervals than the large sample method
• The method works as if four additional observations were made, where two were successes and two were failures
• The new sample size: n + 4, and the count of successes: X + 2
• The "plus four" estimate of p is: p̃ = (count of successes + 2) / (count of all observations + 4)
• The approximate level C confidence interval is CI: p̃ ± m, with m = zŜE = z√p̃(1 – p̃)/(n + 4)
• This method is best employed when C is at least 90% and sample size is at least 10
• Arthritis example with 90% confidence interval for the population proportion of arthritis patients who suffer "adverse symptoms.”
• The value of the “plus four” estimate of p is: p̃ =(23+2)/(440+4)= 25/444 ≈ 0.056
• An approximate 90% confidence interval for p using the plus four method is:
• m = 1.645*√0.056(1 – 0.056)/444 = 0.018
• 90% CI for p: 0.056 ± 0.018.
• Therefore, with 90% confidence, means that between 3.8% and 7.4% of the population of arthritis patients taking the specific pain medication experience adverse symptoms

Choosing the Sample Size

• In some cases, a sample size needs to be chosen to achieve a specified margin of error
• The sampling distribution of p^is a function of the unknown population proportion p; therefore, a likely value for p must be guessed, as p*
• p~N (p,√p(1-p)/n)→n=((z^)/m)^2 p^(1-p^*)
• Make an educated guess, or use p* = 0.5 for the most conservative estimate
• Example
• Need to find the sample size to achieve a margin of error no more than 0.01 (1 percentage point) with a 90% confidence level
• Could use 0.5 for the guessed p*, because the drug has been approved for sale over the counter, it can be assumed less than 10% of patients will suffer symptoms, a better guess than 50%
  
• The 90% confidence level z*= 1.645.
• n=((1.645)/0.01)^2 (0.1)(0.9)=2434.4
• To obtain a margin of error no more than 0.01, need a sample size n of at least 2435 arthritis patients, but using 0.5 for the guess would have resulted in a sample size of 6766 patients

Hypothesis Tests for p

• When testing, Ho: p = po (a given value being tested).
• If Ho is true, the sampling distribution is known.
• The test statistic is the standardized value of p^
• z = (p^-po)/√(po(1-po)/n
• Valid when both expected counts/ successes npo and expected failure n(1-po) are each 10 or larger
• The P-value is the probability, if Ho was true, of obtaining a test statistic like the one computed or more extreme in the direction of Ha
• Ha: p > po is P(Z ≥ z)
• Ha: p < po is P(Z ≤ z)
• Ha: p ≠ po is 2P(Z > |z|)
• Aphid Example
• Live aphids dropped upside-down, landed on their ventral side in 95% of trials versus dead aphids who landed on their ventral side 52.2% of the trials
• Test for evidence (at significance level 5%) to see if live aphids land right side up more often than chance
• Test “chance” to see if it would be 50% ventral landings.
• Test with Ho: p = 0.5 versus Ha: p > 0.5
• z =. (0.95 -0.5)/√(0.5× 0.5)/20≈4.02
• The expected counts of success and failure are each 10, so the z procedure is valid.
• The test P-value is P(z = 4.02). From Table B, P = 1 – P(z < 4.02) < 0.0002 and is highly significant.
• Therefore reject Ho due to strong evidence (P < 0.0002)
• Thus, the righting behavior of live aphids is better than chance.
• Mendels Test
• States that crossing dominant and recessive homozygote parents yields a second generation with 75% dominant traits
• When Mendel crossed pure breeds of plants producing smooth peas and plants producing wrinkled peas, the second generation (F2), was made of 5474 smooth peas and 1850 wrinkled peas
• Test for evidence that the smooth peas in the F2 population is not at 75%
• Test: Ho: p = 0.75 versus Ha: p ≠ 0.75
• The sample proportion is p^ = 5474/ (5474 + 1850)= 0.7474
• z = (0.7474 -0.75)/√(0.75 × 0.25)/7324= -0.513
• From Table B, it is determined that P = 2P(z < -0.51) = 2 × 0.3050 = 0.61 (not significant data)
• Therefore, the claim cannot be rejected because the data are consistent with a dominant-recessive genetic model.

Previous on Biostats: From the exam

• Sample space: A list or description of all possible outcomes of a random process
• An event is a subset of the sample space
• P(A or B) = P(A) + P(B) – P(A and B)
• The probability that a randomly chosen person will test positive depends on True among patients and False among healthy patients
• Concepts regarding parameter vs. statistic are relevant
• Margin of error