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### 104 9.4 conf. int. props.

Course: MATH MATH 2040, Fall 2010
School: Utah Valley University
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Word Count: 428

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Confidence 9.3,9.4 Intervals For Proportions and Testing Hypotheses About Proportions Researchers often want to estimate the proportion of a population with a particular characteristic. For example, A Physical Anthropologist may want to know the fraction of a population with blood type A. To get this estimate the researcher would need to take a random sample of size n, and determine for each individual whether or...

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Confidence 9.3,9.4 Intervals For Proportions and Testing Hypotheses About Proportions Researchers often want to estimate the proportion of a population with a particular characteristic. For example, A Physical Anthropologist may want to know the fraction of a population with blood type A. To get this estimate the researcher would need to take a random sample of size n, and determine for each individual whether or not they have blood type A. For a sample from a large population the number with blood type A satisfies the four requirements of a binomial distribution (assume success is selecting a person with blood type A). 1) Number of trials (individuals sampled) = n 2) Each individual is either blood type A or not type A 3) For a large population there is a constant probability that the next person sampled is type A. 4) the sample is random so the trials are independent. The number in the sample with type A Blood, x, is distributed as a B(n,p) p = type A proportion for the population From chapter seven we know that as the sample size increases a binomial distribution approaches a normal distribution. i.e. B(n,p) N(np, npq ) If we divide x by the sample size we get the sample proportion x/n which also approaches a normal distribution as n increases. Variable mean X np npq p npq n X n standard for deviation = npq = n2 pq n Thus large sample sizes the distribution of the sample proportion, x/n, is close to the normal distribution N(p, pq ) n A confidence interval for the population proportion p can be calculated as for any statistic with a normal distribution. statistic Z / 2 (standard error of the statistic) x n Z / 2 TI calculator A: 1Prop ZInt X: n: C-Level: STAT pq n TESTS Tests Involving Single Proportions Large samples z= A researcher claims the population proportion = .30 Test this claim given the sample data n = 600 and the number in the sample with the characteristic of interest = 157 1) H0: P = .30 Ha: p .30 2) a) = .05 3) Reject H0 if b) = .01 a) |z| 1.96 b) |z| 2.575 4) z = = -2.05 5) a) reject at the 0.05 significance level b) fail to reject at the 0.01 significance level TI calculator STAT 1Prop ZTest P0: .30 X: 157 n: 600 TESTS prop P0 p = .04046 p = .2617 prop P0 9.5 Calculation of the Sample Size We now look at how we determine the sample size needed to estimate p to a specific accuracy and confidence x Let p = n and q =1- p ________________________________________________ pq pq p - Z / 2 p p + Z / 2 n n E= Z / 2 pq n 2 Solving for n Z / 2 n= pq E
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