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9.2a Tests about a Population Proportion

9.2a Tests about a Population Proportion. Target Goal: I can check the conditions for carrying out a test about a population proportion. I can perform a significance test for a sample proportion. h.w : pg 548: 27 – 30, pg 562: 41, 43, 45. Carrying Out a Significance Test.

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9.2a Tests about a Population Proportion

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  1. 9.2a Tests about a Population Proportion Target Goal: I can check the conditions for carrying out a test about a population proportion. I can perform a significance test for a sample proportion. h.w: pg 548: 27 – 30, pg 562: 41, 43, 45

  2. Carrying Out a Significance Test Tests About a Population Proportion Suppose a basketball player who claimed to be an 80% free-throw shooter. In an SRS of 50 free-throws, he made 32. His sample proportion of made shots, 32/50 = 0.64, is much lower than what he claimed. Does it provide convincing evidence against his claim? To find out, we must perform a significance test of H0: p = 0.80 Ha: p < 0.80 where p = the actual proportion of free throws the shooter makes in the long run.

  3. For a hypothesis testwhere Ho: p = po, use poto estimate p. • For a confidence interval: we used as an estimate of .

  4. Assumptions for Inference about a Proportion: 1. Random: SRS 2. Independent: Population  10n 3. Normal:  10 and  10 for a hypothesis test. for a confidence interval.

  5. To test hypothesis Ho: p = po,

  6. The One-Sample z Test for a Proportion The z statistic has approximately the standard Normal distribution when H0is true. P-values therefore come from the standard Normal distribution. Here is a summary of the details for a one-sample z test for a proportion. Tests About a Population Proportion One-Sample z Test for a Proportion Use this test only when the expected numbers of successes and failures np0 and n(1 - p0) are both at least 10 and the population is at least 10 times as large as the sample. Choose an SRS of size n from a large population that contains an unknown proportion p of successes. To test the hypothesis H0: p = p0, compute the z statistic Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the alternative hypothesis Ha: 0

  7. Ex. Binge Drinking in College • In a representative of 140 colleges and 17592 students, 7741 students identify themselves as binge drinkers. Considering this SRS, does this constitute strong evidence that more than 40% of all college students engage in binge drinking?

  8. Step 1: State - What hypotheses do you want to test, and at what significance level? State the hypothesis in words and symbols. • We want to test a claim about the proportion of all U.S. college students who have engaged in binge drinking at the α = .05 level. Our hypotheses are • Ho: p = .4040% of all college students are binge drinkers • Ha: p > .40 more than 40% of all U.S. college students have engaged in binge drinking.

  9. Step 2: Plan - Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. To test claim, we will use the one prop z test . Check conditions. • Random:SRS?We are given SRS. • Independent: Total population > 10 n: 10(17592) = there are more than 175,920 college students in the country. ( to use sample σ) yes independent. • Normal: npo and nqo ≥ 10? 17592(.40) = 7036.8 ≥ 10 17592(.60) = 10,555.2 ≥ 10 Yes, we can use normal approximation.

  10. Step 3:Do - If the conditions are met, carry out the inference procedure. • Calculate z statistic • With a z score this large, the P-value is approx. 0.

  11. Step 4.Conclude - Interpret the results in the context of the problem • The p-value (= 0) this small, < α = .05, tells us that we have no chance of obtaining a sample proportion • We reject Ho and conclude that more than 40% of U.S. college students have engaged in binge drinking.

  12. Ex. Is that Coin Fair? • The French naturalist Count Buffon tossed a coin 4040 times and counted 2048 heads. The sample proportion of heads is • = 0.5069 • Is this evidence that Buffon’s coin was not balanced?

  13. Step 1:State • The parameter p is the probability of tossing a head. The population contains the results of tossing the coin forever. • Our hypotheses are: • Ho: p = .50 The coin is balanced. • Ha: p 0.5 Buffon’s coin is not balanced. The null hypothesis gives p the value po = .50.

  14. Step 2: Plan We will use the one prop z test. Check conditions. • SRS? Yes • Total population > 10 n? • npo and nqo ≥ 10? npo = 4040(0.5) = 2020 ≥ 10 nqo = 4040(0.5) = 2020 ≥ 10 The population of tosses is infinite.

  15. Step 3:Do - If the conditions are met, carry out the inference procedure. • Calculate z statistic • P-value = (two sided so) • Normcdf(.88,E99) 2(.1894) = .3788

  16. Step 4.Conclude - Interpret the results in the context of the problem. • The proportion of heads as far away from 1/2 as Buffon’s would happen 38% of the time. This provides little evidence against Ho. Thus, Buffon’s result doesn’t show that his coin is unbalanced.

  17. Note: calculator much quicker. • STAT: TESTS:1-Prop Z Test • Read pg. 549 - 555

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