Hypothesis Testing Two Proportions

Comparing Two Proportions Comparisons between two percentages are much more common than questions about isolated percentages. And they are more interesting. We often want to know how two groups differ, whether a treatment is better than a placebo control, or whether this year’s results are better than last year’s.

Another Ruler In order to examine the difference between two proportions, we need another ruler—the standard deviation of the sampling distribution model for the difference between two proportions. Recall that standard deviations don’t add, but variances do. In fact, the variance of the sum or difference of two independent random variables is the sum of their individual variances.

The Standard Deviation of the Difference Between Two Proportions Proportions observed in independent random samples are independent. Thus, we can add their variances. So… The standard deviation of the difference between two sample proportions is Thus, the standard error is

Assumptions and Conditions Independence Assumptions: Randomization Condition: The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative experiment. The 10% Condition: If the data are sampled without replacement, the sample should not exceed 10% of the population. Independent Groups Assumption: The two groups we’re comparing must be independent of each other.

Assumptions and Conditions (cont.) Sample Size Condition: Each of the groups must be big enough… Success/Failure Condition: Both groups are big enough that at least 10 successes and at least 10 failures have been observed in each. CONDITIONS ARE EXACTLY THE SAME AS A ONE PROPORTION TEST! You just have to check them for each sample.

The Sampling Distribution We already know that for large enough samples, each of our proportions has an approximately Normal sampling distribution. The same is true of their difference.

The Sampling Distribution (cont.) Provided that the sampled values are independent, the samples are independent, and the samples sizes are large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:

Everyone into the Pool ALWAYS POOL DATA ON MINITAB!!!! The typical hypothesis test for the difference in two proportions is the one of no difference. In symbols, H0: p1 – p2 = 0. Since we are hypothesizing that there is no difference between the two proportions, that means that the standard deviations for each proportion are the same. Since this is the case, we combine (pool) the counts to get one overall proportion. ALWAYS POOL DATA ON MINITAB!!!!

Suppose the Acme Drug Company develops a new drug, designed to prevent colds. The company states that the drug is equally effective for men and women. To test this claim, they choose a simple random sample of 100 women and 200 men from a population of 100,000 volunteers. At the end of the study, 38% of the women caught a cold; and 51% of the men caught a cold. Based on these findings, can we reject the company's claim that the drug is equally effective for men and women?

Understanding our Conclusions P-value: The probability we would observe a statistic this extreme given the null hypothesis is true. Small p-values (less than 5%) we REJECT the null hypothesis. Larger p-values (greater than 5%) we FAIL TO REJECT the null hypothesis.

Right or Wrong? If we reject or fail to reject does it mean we made the right decision? Nobody’s perfect. Even with lots of evidence we can still make the wrong decision.

Making Errors Here’s some shocking news for you: nobody’s perfect. Even with lots of evidence we can still make the wrong decision. When we perform a hypothesis test, we can make mistakes in two ways: The null hypothesis is true, but we mistakenly reject it. (Type I error) The null hypothesis is false, but we fail to reject it. (Type II error)

Making Errors (cont.) Which type of error is more serious depends on the situation at hand. In other words, the gravity of the error is context dependent. Here’s an illustration of the four situations in a hypothesis test:

Examples Ho: There is no difference in the new medicine. Ha: The new medicine works better. A company is going to spend millions of dollars to mass produce this new drug if they feel it works better.

Examples Ho: You won’t win the lottery. Ha: You will win the lottery. If you don’t think you will win you won’t buy a ticket, if you think you win you will buy a $2 ticket.