Math215 – Statistical Concepts Hypothesis Testing Group Worksheet

First review the worked examples and then solve the problems….e your TI calculator to test a hypothesis of a mean or proportion. Study the workedexamples below, understand the logic of hypothesis testing and learn the calculator keystrokes. Then work on the problems that follow.1. Hypothesis Testing of a mean, µ, when ? is known:Technical University did a study in 2007 regarding the number of hours full-timestudents worked each week. The mean number of hours was 8.7. A recent studyinvolving a random sample of 65 full-time students showed that the average number ofhours worked per week was 10.3. Also assume that the population standard deviationwas 2.8. Use a 5% level of significance to test if the mean number of hours worked perweek by full-time students has increased since 2007.State the null and alternative hypothesis:H 0 : ?=8.7H 1 : ?> 8.7Because the population standard deviation, ?, is known this is a Z-Test. To access the ZTest, press STAT and select TESTS. Select 1:Z-Test . This example gives summarystatistics, so select the STATS option for input. Enter the population mean from the nullhypothesis, the value of ?, the sample mean, ´x , and the sample size n. Choose?> ?0 to indicate that the alternative is that µ > 8.7. Either select Calculate to readthe test statistic and p-value or Draw which also gives both values and show the teststatistic on the normal curve. Because the sample z is so far to the right, it does notappear on this window. However, its value does, and a rounded p- value shows as well.On the curve the area of the p-value will be shaded. nNewdaRcylWfrs(thm). 1The results show that the zvalue corresponding to thesample test statistic ´x =10.3, is z = 4.61. This is the teststatistic. Notice that the p -value = 2.0446899E-6. The -6 at the end of the number means tomove the decimal 6 places to the left, giving a p- value of 0.000002. Since the p -value isless than 0.05, the decision is to reject H0 . The evidence for the alternative hypothesiswas strong so the evidence suggests that the number of hours worked by full timestudents has increased.2. Hypothesis Testing of a mean, µ, when ? is unknown:A new postal sorting machine was advertised as being able to sort a mean of 110 piecesof mail every minute. Twenty one-minute segments were randomly selected and thenumber pieces of mail sorted was recorded for each. The twenty segments had a meanof 109 pieces of mail with a standard deviation of 3 pieces. Assume that this distributionis relatively normal and test if the machine is processing fewer pieces of mail thanadvertised. (use ? = .05)First identify the null and alternative hypotheses.H 0 : ?=110H 1 : ?<110 .Since the standard deviation of the population, ? , is not given here, the zdistribution can’t be used. Use the t-distribution. To access T-Test, press STAT and selectTESTS. Select 2:T-Test. In this example, summary statistics are given, so select the STATSoption for input. Enter the population mean, µ as stated in the null, the sample mean,´x , the standard deviation of the sample, s, and the sample size n. Choose ?< ?0 toindicate that the alternative hypothesis is ?<110 . Either select Calculate to read thetest statistic and P-value or Draw which also gives both values and shows the teststatistic on the normal curve. New and RcylttWfhrs(mt).Since P-value given is 0.076 and it is greater than the .05 level of significance thedecision is to fail to reject the null hypothesis and conclude that there is not sufficientevidence to suggest that mail sorting machine is not working according to the publishedstandard.3. Hypothesis testing for a population proportion:In a poll of 745 randomly selected adults, 589 said that it is morally wrong to notreport all income on tax returns. Use a 0.01 significance level to test the claim that 75%of all adults say that it is morally wrong to not report all income on tax returns.The point estimate is 589745 = 0.79. The null and alternative hypotheses are: H0 : p= 0.75H1: p > 0.75Notice that in this problem the parameter is p in the hypothesis. This shows thatthis is a problem testing a population proportion. To conduct a hypotheses test of asingle proportion, select option 5:1-PropZTest. The number of successes is designated bythe value x, which is 589 in this example. Enter the value. The sample size or number oftrials is n, which is 745. The alternate hypothesis will be prop ? p 0, < p0, or > p0, where thep0 is the value stated in the null hypothesis. (Here p0 is .75.) Highlight the appropriatechoice. Finally highlight Calculate and press ENTER. The Draw option is also available toshow the results on the standard normal distribution. Since the P-value is 0.005 which is less than the significance level of 0.01, the decision isto reject the null hypothesis. There is strong evidence to say that more than 75% ofadults agree that it is morally wrong to not report all income on tax returns.For the following exercises, use your calculator to conduct the hypothesis test.Remember to use the correct test on the calculator (Are you testing a mean or aproportion? Is ? given or not?) Questions1. In a survey of 688 US adults with children, 249 of them reported that they savedmoney for their children’s college education. Can you conclude that more than onethird (33%) of US adults with children have saved money for college? Use a 0.05significance level.a. State the null (H0) and alternative (H1) hypotheses. b. Give the test statistics and the p-value for this significance test. c. Make a decision on whether or not to reject the null hypothesis. d. Summarize the conclusion in the context of this problem. 2. The mean IQ for the entire population in any age group is supposed to be 100.Suppose the IQ scores of 72 10 th graders in Columbus are measured and the samplemean is found to be 105.4 and the sample standard deviation is 21.2. Do the scoresprovide good evidence that the mean IQ of this population is more than 100?a. State the null (H0) and alternative (H1) hypotheses.b. Give the test statistics and the p-value for this significance test. c. Make a decision on whether or not to reject the null hypothesis.d. Summarize the conclusion in the context of this problem.