Section 9.4 Hypothesis Tests for Proportions 449 EXAMPLE 9.19 Perform a hypothesis test In a recent GenX2Z American College Student Survey, 90% of female college students rated the social network site Facebook as ‘‘cool.’’ Assume that the survey was based on a random sample of 500 students. A marketing executive at Facebook wants to advertise the site with the slogan ‘‘More than 85% of female college students think Facebook is cool.’’ Can you conclude that the proportion of female college students who think Facebook is cool is greater than 0.85? Use the �� = 0.05 level of significance. Solution We first check the assumptions. We have a simple random sample of students. The members of the population fall into two categories: those who think that Facebook is cool and those who don’t. The size of the population of female college students is more than 20 times the sample size of n = 500. The proportion specified by the null hypothesis is p0 = 0.85. Now np0 = (500)(0.85) = 425 > 10 and n(1 − p0) = (500)(1 − 0.85) = 75 > 10. The assumptions are satisfied. Step 1: State H0 and H1. We are asked whether we can conclude that the population proportion p is greater than 0.85. The null and alternate hypotheses are therefore H0 : p = 0.85 H1: p > 0.85 Step 2: Choose a significance level. The significance level is �� = 0.05. Step 3: Compute the test statistic. The sample proportion ̂p is 0.90. The value of p specified by the null hypothesis is p0 = 0.85. The test statistic is the z-score for ̂p: z = ̂p − p0 √ p0(1 − p0) n = 0.90 − 0.85 √ 0.85(1 − 0.85) 500 = 3.13 Step 4: Compute the P-value. The alternate hypothesis is H1: �� > 0.85, which is righttailed. The P-value is therefore the area to the right of z = 3.13. Using Table A.2, we see that the area to the left of z = 3.13 is 0.9991. The area to the right of z = 3.13 is therefore 1 − 0.9991 = 0.0009. The P-value is P = 0.0009. See Figure 9.16. Area = 0.0009 z = 3.13 Figure 9.16 Step 5: Interpret the P-value. A P-value of P = 0.0009 is very small. This is very strong evidence against H0. In particular, because P < 0.05, we reject H0 at the �� = 0.05 level. Step 6: State a conclusion. We conclude that more than 85% of female college students think Facebook is cool. Check Your Understanding 1. The Pew Research Center reported that only 15% of 18- to 24-year-olds read a daily newspaper. The publisher of a local newspaper wants to know whether the percentage of newspaper readers among students at a nearby large university differs from the percentage among 18- to 24-year-olds in general. She surveys a simple random
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