# Away from the true proportion how many senior

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away from the true proportion. How many senior citizens do we need for our study? Example: Suppose we think that the 1953 proportion might be signifi- cantly different from the current proportion. What is the largest sample size we’ll need to guarantee that the confidence interval will be no more than 0 . 03 away from the true proportion? 21
Hypothesis tests on population proportion: We use the following as our test statistic. z = ˆ p - p r p (1 - p ) n Note: Be careful! In the confidence interval, we used the estimated standard error p ˆ p (1 - ˆ p ) /n since we did not have a value of p to use. For hypothesis tests, since we do have a hypothesized value of p , we use the standard error p p (1 - p ) /n . 22
Example: At a hospital, the administrators have decided that the ac- ceptable proportion of correctly interpreted scans must be 98% or better. In a random sample of 500 scans, it is found that 487 were interpreted correctly. Is there evidence that the proportion of correctly interpreted scans is below the acceptable level? Test your hypothesis at the level α = 0 . 15 . 23
Sets 24 Instead of drawing samples from one population, we may take random samples from two populations so that we can carry out some sort of comparison. We may wish to compare p 1 and p 2 , the population proportions for pop- ulations 1 and 2 . We do so by examining the difference, p 1 - p 2 . Example: Suppose we wish to compare the p 1 , the proportion of people in BC with diabetes, with p 2 , the proportion of people in Alberta with diabetes. If the proportions are equal, then p 1 - p 2 = 0 . If the proportions are different, then p 1 - p 2 6 = 0 . If the proportion is higher in BC, then p 1 > p 2 , which means that p 1 - p 2 > 0 . If the proportion is higher in Alberta, then p 1 < p 2 , which means that p 1 - p 2 < 0 . If the proportion in BC is higher than in Alberta by at least 0 . 05 , then p 1 > p 2 + 0 . 05 , which means that p 1 - p 2 > 0 . 05 If the proportion in Alberta is higher than in BC by at least 0 . 02 , then p 1 + 0 . 02 < p 2 , which means that p 1 - p 2 < - 0 . 02 24
We use ˆ p 1 - ˆ p 2 as the point estimate for p 1 - p 2 Distribution of ˆ p 1 - ˆ p 2 : Suppose we have n 1 samples from a first population, and n 2 samples from a second population. 1. ˆ p 1 - ˆ p 2 has a normal distribution. 2. ˆ p 1 - ˆ p 2 has mean p 1 - p 2 and has standard error: s p 1 (1 - p 1 ) n 1 + p 2 (1 - p 2 ) n 2 25
Confidence interval for difference in population proportions: ( ˆ p 1 - ˆ p 2 ) ± z α/ 2 · s ˆ p 1 (1 - ˆ p 1 ) n 1 + ˆ p 2 (1 - ˆ p 2 ) n 2