Inference for Two Proportions Chapter 22

Conditions: Two, independent SRS’s from populations Each sample <10% of Population 10 successes and failures for each

Formula for confidence interval: Margin of error! Standard error!

Example 1: At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. For this group, it was found that 94 had no visible scars after treatment. What is the shape & standard error of the sampling distribution of the difference in the proportions of people with visible scars between the two groups? Since n1p1=259, n1(1-p1)=57, n2p2=94, n2(1-p2)=325 and all > 10, then the sampling distribution of difference in proportions is approximately normal.

Example 1: At Community Hospital, the burn center is experimenting with a new plasma compress treatment. A random sample of 316 patients with minor burns received the plasma compress treatment. Of these patients, it was found that 259 had no visible scars after treatment. Another random sample of 419 patients with minor burns received no plasma compress treatment. For this group, it was found that 94 had no visible scars after treatment. What is a 95% confidence interval of the difference in proportion of people who have no visible scars after having the plasma compress treatment versus no treatment?

p1 = true proportion of patients who received the plasma treatment and showed no visible scars p2 = true proportion of patients who did not receive the plasma treatment and showed no visible scars Conditions: We have 2 random samples of burn patients It is reasonable to assume that the samples are independent Assume both 316 patients and 419 patients are <10% of their respective populations. The sampling distribution of the difference in proportions is approximately normal since

We are 95% confident that the true difference in the proportion of people who have no visible scars after the plasma compress treatment versus no treatment is between 53.7% and 65.4% We are 95% confident that between 53.7% and 65.4% more patients who receive the plasma compress will have no visible scars than patients who receive no treatment.

SO – which is correct? CIA = (.67, .83) CIB =(.52, .70) Example 2: Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, and then administered the painkillers. Of the 112 people in the group who received medication A, 84 said this pain reliever was effective. Of the 108 people in the other group, 66 reported that pain reliever B was effective. a) Construct separate 95% confidence intervals for the proportion of people who reported that the pain reliever was effective. Based on these intervals how do the proportions of people who reported pain relieve with medication A or medication B compare? b) Construct a 95% confidence interval for the difference in the proportions of people who may find these medications effective. SO – which is correct? CIA = (.67, .83) CIB =(.52, .70) Since the intervals overlap, it appears that there is no difference in the proportion of people who reported pain relieve between the two medicines. CI = (0.017, 0.261) Since zero is not in the 95% confidence interval, there is evidence of a difference in the proportion of people who reported pain relief between the two medicines.

Example 3: Suppose that researchers want to estimate the difference in proportions of people who are against the death penalty in Texas & in California. If the two sample sizes are the same, what size sample is needed to be within 2% of the true difference at 90% confidence? Since both n’s are the same size, you have common denominators – so add! n = 3383

Hypothesis statements: H0: p1 = p2 or H0: p1 - p2 = 0 HA: p1 > p2 or HA: p1 - p2 > 0 HA: p1 < p2 or HA: p1 - p2 < 0 HA: p1 ≠ p2 or HA: p1 - p2 ≠ 0 Be sure to define both p1 & p2!

Since we assume that the population proportions are equal in the null hypothesis, the variances are equal. Therefore, we pool the variances!

Formula for Hypothesis test: p1 = p2 So . . . p1 – p2 =0

Example 4: A forest in Oregon has an infestation of spruce moths Example 4: A forest in Oregon has an infestation of spruce moths. In an effort to control the moth, one area has been regularly sprayed from airplanes. In this area, a random sample of 495 spruce trees showed that 81 had been killed by moths. A second nearby area receives no treatment. In this area, a random sample of 518 spruce trees showed that 92 had been killed by the moth. Do these data indicate that the proportion of spruce trees killed by the moth is different for these areas?

p1 = the true proportion of trees killed by moths in the treated area p2 = the true proportion of trees killed by moths in the untreated area H0: p1-p2=0 HA: p1-p2≠0 Conditions: Have 2 random samples The 2 samples are independent We assume both samples are <10% of each population. The sampling distribution of the difference in proportions is approximately normal since

P-value = 0.5547 a = 0.05 Since my p-value > a, I fail to reject H0. There is not sufficient evidence to suggest that the proportion of spruce trees killed by the moth is different between the treated and untreated areas

Example 5: A Gallup Poll taken in May 2000 asked the question: “In general, do you feel that the laws covering the sale of firearms should be made: more strict, less strict, or kept as they are now?” Of the 493 men who responded, 52% said “more strict,” while of the 538 women who responded, 72% said “more strict.” Assuming these respondents constitute random samples of U.S. men and women, is there sufficient evidence to conclude that a higher proportion of women than men in the population think these laws should be made more strict? Justify your answer.