Hypothesis Tests for Proportions Section 9.4

Objectives Test a hypothesis about a proportion using the P-value method Test a hypothesis about a proportion using the critical value method

Test a hypothesis about a proportion using the P-value method Objective 1 Test a hypothesis about a proportion using the P-value method

Introduction In a recent GenX2Z American College Student Survey, 90% of female college students rated the social network site Facebook as “cool.” The other 10% rated it as “lame.” Assume that the survey was based on a 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.” Before launching the ad campaign, he wants to be confident that the slogan is true. Can he conclude that the proportion of female college students who think Facebook is cool is greater than 0.85? This is an example of a problem that calls for a hypothesis test about a population proportion.

Notation We use the following notation: 𝑝 is the population proportion of individuals who are in a specified category. 𝑥 is the number of individuals in the sample who are in the specified category. 𝑛 is the sample size. 𝑝 is the sample proportion of individuals who are in the specified category. 𝑝 is defined as 𝑝 = 𝑥 𝑛 .

Assumptions The method for performing a hypothesis test about a population proportion requires that the sampling distribution be approximately normal. The following assumptions ensure this: Assumptions: We have a simple random sample. The population is at least 20 times as large as the sample. The items in the population are divided into two categories. The values 𝑛 𝑝 0 and 𝑛(1 − 𝑝 0 ) are both at least 10.

P-Value Method Step 1: State the null and alternate hypotheses. Step 2: If making a decision, choose a significance level 𝛼. Step 3: Compute the test statistic 𝑧= 𝑝 − 𝑝 0 𝑝 0 1− 𝑝 0 𝑛 . Step 4: Compute the P-value. Step 5: Interpret the P-value. If making a decision, reject 𝐻 0 if the P-value is less than or equal to the significance level 𝛼. Step 6: State a conclusion.

Example 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 𝛼 = 0.05 level of significance. Solution: 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 𝑝 0 = 0.85. Now 𝑛 𝑝 0 = (500)(0.85) = 425 > 10 and 𝑛 1− 𝑝 0 = (500)(1 − 0.85) = 75 > 10. The assumptions are satisfied. The null and alternate hypotheses are: 𝐻 0 :𝑝=0.85 versus 𝐻 1 :𝑝>0.85

Example Solution (continued): The sample proportion 𝑝 is 0.90. The value of 𝑝 specified by the null hypothesis is 𝑝 0 = 0.85. The test statistic is the 𝑧-score of 𝑝 and is given by: 𝑧= 𝑝 − 𝑝 0 𝑝 0 1− 𝑝 0 𝑛 = 0.90 − 0.85 0.85 1 − 0.85 500 =3.13 This is a right-tailed test, so the P-value is the area to the right of 𝑧 = 3.13. We may use Table A.2 or technology to find that this is 0.0009. Since P < 0.05, we reject 𝐻 0 at the 𝛼 = 0.05 level and conclude that more than 85% of female college students think that Facebook is cool. Remember 𝐻 0 :𝑝=0.85 𝐻 1 :𝑝>0.85 𝑛=500 𝑝 =0.90

Objective 1 Test a hypothesis about a proportion using the P-value method (TI-84 PLUS)

Introduction In a recent GenX2Z American College Student Survey, 90% of female college students rated the social network site Facebook as “cool.” The other 10% rated it as “lame.” Assume that the survey was based on a 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.” Before launching the ad campaign, he wants to be confident that the slogan is true. Can he conclude that the proportion of female college students who think Facebook is cool is greater than 0.85? This is an example of a problem that calls for a hypothesis test about a population proportion.

Notation We use the following notation: 𝑝 is the population proportion of individuals who are in a specified category. 𝑥 is the number of individuals in the sample who are in the specified category. 𝑛 is the sample size. 𝑝 is the sample proportion of individuals who are in the specified category. 𝑝 is defined as 𝑝 = 𝑥 𝑛 .

Assumptions The method for performing a hypothesis test about a population proportion requires that the sampling distribution be approximately normal. The following assumptions ensure this: Assumptions: We have a simple random sample. The population is at least 20 times as large as the sample. The items in the population are divided into two categories. The values 𝑛 𝑝 0 and 𝑛(1 − 𝑝 0 ) are both at least 10.

Hypothesis Testing on the TI-84 PLUS The 1-PropZTest command will perform a hypothesis test for a population proportion. This command is accessed by pressing STAT and highlighting the TESTS menu. The required inputs for the 1-PropZTest are the values of 𝑥 and 𝑛. If the sample proportion 𝑝 is given in the problem, the value of 𝑥 can be computed as 𝑥 = 𝑝 ·𝑛.

Example 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 𝛼 = 0.05 level of significance. Solution: 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 𝑝 0 = 0.85. Now 𝑛 𝑝 0 = (500)(0.85) = 425 > 10 and 𝑛 1− 𝑝 0 = (500)(1 − 0.85) = 75 > 10. The assumptions are satisfied. The null and alternate hypotheses are: 𝐻 0 :𝑝=0.85 versus 𝐻 1 :𝑝>0.85

Example (TI-84 PLUS) Remember 𝐻 0 :𝑝=0.85 Solution (continued): We press STAT and highlight the TESTS menu and select 1-PropZTest. We are given the sample proportion 𝑝 = 0.90 and 𝑛 = 500. We compute the value of 𝑥 as 𝑥 = 𝑝 ·𝑛 = 0.90(500) = 450. We enter 𝑥 = 450 and 𝑛 = 500. Since this is a right-tailed test, we select > 𝒑 𝟎 . Select Calculate. The P-value is approximately 0.0009. Since P < 0.05, we reject 𝐻 0 at the 𝛼 = 0.05 level and conclude that more than 85% of female college students think that Facebook is cool. Remember 𝐻 0 :𝑝=0.85 𝐻 1 :𝑝>0.85 𝑛=500 𝑝 =0.90

Test a hypothesis about a proportion using the critical value method Objective 2 Test a hypothesis about a proportion using the critical value method

Critical Value Method Step 1. State the null and alternate hypotheses. Step 2. Choose a significance level 𝛼 and find the critical value(s). Step 3. Compute the test statistic 𝑧= 𝑝 − 𝑝 0 𝑝 0 1− 𝑝 0 𝑛 . Step 4. Determine whether to reject 𝐻 0 as follows: Step 5. State a conclusion.

Example A nationwide survey of working adults indicates that only 50% of them are satisfied with their jobs. The president of a large company believes that more than 50% of employees at his company are satisfied with their jobs. To test his belief, he surveys a random sample of 100 employees, and 54 of them report that they are satisfied with their jobs. Can he conclude that more than 50% of employees at the company are satisfied with their jobs? Use the 𝛼 = 0.05 level of significance. Solution: We have a simple random sample from the population of employees. Each employee is categorized as being satisfied or not satisfied. The sample size is 𝑛 = 100 and the proportion 𝑝 0 specified by 𝐻 0 is 0.5. Therefore 𝑛 𝑝 0 = 100(0.5) = 50 > 10, and n(1 − 𝑝 0 ) = 100(1 − 0.5) = 50 > 10. If the total number of employees in the company is more than 2000, as we shall assume, then the population is more than 20 times as large as the sample. All the assumptions are therefore satisfied.

Example Solution (continued): The null and alternate hypotheses are: 𝐻 0 :𝑝=0.5 versus 𝐻 1 :𝑝>0.5 Since the alternate hypothesis is 𝑝 > 0.5, this is a right-tailed test. The critical value is 𝑧 𝛼 = 1.645. Recall that 𝑝 = 𝑥 𝑛 = 54 100 =0.54, thus 𝑧= 𝑝 − 𝑝 0 𝑝 0 1− 𝑝 0 𝑛 = 0.54 −0.5 0.5(1 −0.5) 100 =0.80. Because this is a right-tailed test, we reject 𝐻 0 if 𝑧 ≥1.645. Because 0.80 < 1.645, we do not reject 𝐻 0 . There is not enough evidence to conclude that company president is correct in his belief that the proportion of employees who are satisfied with their jobs is greater than 0.5. Remember 𝑥=54 𝑛=100 𝑝 0 =0.5

You Should Know… The notations used in performing a hypothesis test about a population proportion The assumptions for performing a hypothesis test about a population proportion How to perform a hypothesis test about a population proportion using the P-value method How to perform a hypothesis test about a population proportion using the critical value method