9.6a Developing Hypothesis and understanding Possible Conclusions

An airline company claims in a recent advertisement that more than 94% of passenger luggage that is lost is recovered and reunited with the customer within 1 day. Hunter is a graduate student studying statistics. For a research project, Hunter wants to find out whether there is convincing evidence in support of the airline company's claim. He randomly selects 315 passengers of the airline whose luggage was lost by the airline and found that 276 of those passengers were reunited with their luggage within 1 day. Are all of the conditions for this hypothesis test met, and if so, what are the null and alternative hypotheses for this hypothesis test?

All of the conditions to conduct the hypothesis test have been met. The null and alternative hypotheses are H0:p=0.94 Ha:p>0.94. First verify whether all of the conditions have been met. Let p be the population proportion for the airline passengers whose luggage was lost by the airline and were reunited with their luggage within 1 day. Since Hunter is completing a survey where there are two independent outcomes, the proportion follows a binomial model. The question states that Hunter randomly selected the airline passengers whose luggage was lost by the airline. The expected number of successes, np=296.1, and the expected number of failures, nq=n(1−p)=18.9, are both greater than or equal to 5. Since all of the conditions for this hypothesis test have been satisfied, determine the null and alternative hypotheses. Since Hunter is determining whether the proportion for reuniting passengers with their luggage within 1 day is greater than 94%, the null hypothesis is that p is equal to 0.94 and the alternative hypothesis is that p is greater than 0.94. The null and alternative hypotheses are shown below. H0:p=0.94 Ha:p>0.94.

Colin is a student in a statistics course looking to show whether the proportion of games played that are won by the visiting team in a certain professional lacrosse league is 50%. Colin randomly selects 83 games from the past few seasons and finds that 37 of the games were won by the visiting team. What are the null and alternative hypotheses for this hypothesis test?

H0:p=0.5 Ha:p≠0.5 First verify whether all of the conditions have been met. Let p be the population proportion for the visiting team winning a game played in the league. 1. Since there are two independent outcomes for each trial, the proportion follows a binomial model. 2. The question states that the sample was collected randomly. 3. The expected number of successes, np=41.5, and the expected number of failures, nq=n(1−p)=41.5, are both greater than or equal to 5. Since Colin is testing whether 50% of the games played in the league are won by the visiting team, the null hypothesis is that p is equal to 0.5 and the alternative hypothesis is that p is not equal to 0.5. The null and alternative hypotheses are shown below.

Austin works in the marketing department for a large cable company. Recently, he heard that 56% of cable TV subscribers are considering dropping their cable TV subscription. He tests the claim by randomly selecting a sample of cable TV subscribers and asking whether they are considering dropping their cable TV subscription. Austin finds that 122 of the 224 he surveyed are considering dropping their cable TV subscription. What are the null and alternative hypotheses for this hypothesis test?

H0:p=0.56 Ha:p≠0.56 First verify whether all of the conditions have been met. Let p be the population proportion for cable TV subscribers who are considering dropping their cable TV subscription. Since there are two independent outcomes for each trial, the proportion follows a binomial model. The question states that the sample was collected randomly. The expected number of successes, np=125.44, and the expected number of failures, nq=n(1−p)=98.56, are both greater than or equal to 5. Since Austin is looking for evidence that supports the claim that 56% of cable TV subscribers are considering dropping their cable TV subscription, the null hypothesis is that p is equal to 0.56 and the alternative hypothesis is that p is not equal to 0.56. The null and alternative hypotheses are shown below. H0:p=0.56 Ha:p≠0.56

Kylie works for a large nursery and is investigating whether to use a new brand of seeds. The new brand of seeds advertises that 93% of the seeds germinate, which is higher than the germination rate of the seeds she is currently using. She will change over to this new brand unless the actual germination rate is less than what is advertised. Kylie conducts an experiment by randomly selecting 76 seeds of the new brand and plants them. She finds that 70 of those seeds germinated. What are the null and alternative hypotheses for this hypothesis test?

H0:p=0.93 Ha:p<0.93 First verify whether all of the conditions have been met. Let p be the population proportion for the germination rate of the new seeds. 1.Since there are two independent outcomes for each trial, the proportion follows a binomial model. 2. The question states that the sample was collected randomly. 3. The expected number of successes, np=70.68, and the expected number of failures, nq=n(1−p)=5.32, are both greater than or equal to 5. Since Kylie is testing whether the germination rate is less than 93%, the null hypothesis is that p is equal to 0.93 and the alternative hypothesis is that p is less than 0.93. The null and alternative hypotheses are shown below.