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Problem 3 Medium Difficulty

For each situation described below, identify the population and the sample, explain what $p$ and $\hat{p}$ represent, and tell whether the methods of this chapter can be used to create a confidence interval.
a) Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of the 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe.
b) A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 602 people who voted, 488 favored prayer in schools. We want to estimate the level of support among the general public.
c) A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 1245 students; 380 surveys are returned, with 228 families in favor of the change.
d) A college admits 1632 freshmen one year, and four years later 1388 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

Answer

a) If the sample data is representative then, we can apply the methods which are discussed in this
chapter.
b) Therefore the sample data is biased and non-random. So we can not apply the methods which are discussed in this chapter.
c) Questionnaire is send to 1245 home but only 380 surveys are returned, remaining 865 questionnaire surveys are not returned. So there is a non-response bias. Therefore, we use caution with methods of this chapter.
d) If that year's students are viewed as a representative sample of all possible students at the school then, we can use the methods which are discussed in this chapter.

Discussion

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Video Transcript

this question were asked to identify the population and sample based on different scenarios and tell whether or not they can be used to create a confidence interval. So let's first start off by just defining the variables. So we have P, which is the population proportion and P hot, which is the symptom, the sample proportion. So in taste A our population will be all the cars, whereas the sample size is going to be the cars stopped at the certain checkpoints and like we said, p as the population proportion. So in this case, is all cars with safety problems, and P hot is a sample proportion. So these air the cars that are actually seen with safety problems. So we can further calculate P hat by using the numbers given. And we know that there are 14 of 134 cars stopped have at least one safety problem, so that number ends up being 0.1045 Weaken further transfer that into percentage form and we get that it's 10 points 45% as RP hot volume and were also asked whether these methods can be used to create the confidence interval. So when a sample of data is representative, then it can be used to create a confidence interval and in this case it is because it's sampling all cars. For case be, we are going to find the population and sample once again for the population. We have the general public and for the sample, it's people that are logged into the website. We can further define them as P being the favor. The people in favor of prayer in school where us The sample proportion are the people that voted in this poll who favor prayer in school, we can calculate p hot with the given values were at 488 over 602. We get 0.81 and making that into a percentage value, we get 81% toe. Decide whether or not the sample can be used. We can Onley consider people logging into the website for this case. So in a way it's a bit biased and non random. So you're unable to apply the methods to create the confidence interval in Casey. The population is the parents at school and the sample is the parents expressing opinions through the question here. So the population proportion are all parents who favor the uniforms, whereas the sample proportion P hat are the respondents favoring uniforms. We can calculate the P hot value based on the numbers given 228 over 380 and that gives us a value of 0.6 that can be converted into a percentage of 60%. And since there were 1245 surveys sent home but only 380 returned, there is a complication of non response bias. And so you would use these methods with caution if creating a confidence interval. And the last part D were given a population of students at college, and the sample size is the 16 31,632 College admits the population proportion are all the students who will graduate on time and P hot the sample proportion as the students graduating on time that year. So based on the given values, we have 1388 over 632. Actually, this number is supposed to be 16 32. Sorry, and so based on that we get a value of 0.85 and that could be converted to 85% based on this value, and the sample data for this case was pretty representative. So since it is representative, you can apply these methods to create a confidence interval.