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Numerade Educator

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Problem 8 Hard Difficulty

One year, there were about $3,000$ institutions of higher learning in the U.S. (including junior colleges and community colleges). As part of a continuing study of higher education, the Carnegit Commission took a simple random sample of 400 of these institutions.? The average enrollment in the 400 sample schools was $3,700,$ and the SD was $6,500 .$ The Commission estimates the average enrollment at all $3,000$ institutions to be around $3,700 ;$ they put a give-or-take number of 325 on this estimate. Say whether each of the following statements is true or false, and explain. If you need more information to decide, say what you need and why.
(a) An approximate 68$\%$ -confidence interval for the average enrollment of all 3, 000 institutions from $3,375$ to $4,025 .$
(b) If a statistician takes a simple random sample of 400 institutions out of $3,000,$ and goes one SE either way from the average enrollment of the 400 sample schools, there is about a 68$\%$ chance that his interval will cover the average enrollment of all $3,000$ schools.
(c) About 68$\%$ of the schools in the sample had enrollments in the range $3,700 \pm 6,500 .$
(d) It is estimated that 68$\%$ of the $3,000$ institutions of higher learning in the U.S. enrolled between $3,700-325=3,375$ and $3,700+325=$ $4,025$ students.
(e) The normal curve can't be used to figure confidence levels here at all, because the data don't follow the normal curve.

Answer

a) true.
b) true.
c) false.
d) true.
e) false.

Discussion

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Video Transcript

and this particular question we have given us that there are 3000 institutions and we have taken a sample of $400 per foot. So are the leader has been given to us. So I want to write it down here. The sample size and is 400 schools and, uh, the average in Norman him. Take expert is giving us 3700 then Understanding division for the sample is also given me just six powers of 500. And, uh, the commission has put a give or take off standard error, so that standard at all is given as 3 25 on the estimate. So I'm going to start with the first part here. The question say's that for the first part, they have asked that an approximate 68 core person of confidence in double for the average enrollment off all 3000 institutions from 3375 to 4 to five. So meaning to check the statement is true or false. So if I start with the explanation now, I want to see that, uh, the what have you done is that the 60 person off the glass and debris for the average in government. What? Steven, it wasn't a question for the average enrollment off all the 3000 institutions. Eyes running from him saying 3375 tau four or five. So therefore, I can see that the interval that is given to us, which is ah, 3375 to 4025 is one standard editor away from the average in government. Yeah, So I can therefore say that one standard error. Oh, God. Response to 68% off confidence and double in the normal level. So we can therefore see that the statement is don't now moving on to the next part for but me. The question states that if the status institution takes a simple random sample of 400 institutions out of 2000 and goes on a standard error either way from the average enrollment off, wondered sample schools still be about 60% chance off the double, which will cover the everyday enrollment off all 3000 schools. So he's supposed to check if this is a true or false statement, so I can see that it is already Norm you found in party that 60% off samples cover the population average off endorsement from 3375 to 4 or five. So therefore, control that the statement is true. Someone who already learned that from party it is known waas that 68% off all samples cover the population average off a Norman from 3375 to 4025. So therefore, I will say that is given statement for the parties. Also true moving onto the bar. See, the question states that about 60% off the schools in the sample had enrollment in this change off 2700 plus or minus 6500. So you need to check if this is a true or false statement. So if I start with the park, See, uh, we can say again that you know, it's already been proved here and Barbie again that 68% off schools in the sample had enrollments in the range. You can see it's a strong 23375240 Goofier. So that means the statement of what they've given his force. So we therefore say that, uh, the Steve Moon that a boat. 68% with schools in the sample had enrollments in the range 3700 plus or minus 6500. So we say that this statement is false because we've already proved earlier that it is from 2375 to 4025. Now, will you go on to the next part the butt d for the Barbie, The question states that it is estimated that 68% off the 3000 institutions off higher learning in the U. S. S and all between 50,007 dead minds Treat my friends, which is 3375 and it's 2700 plus t 25. So which is for water Firestones? So, uh, you want to prove this part in but a and that is a double order given to us. So we need to again check on dhe side effects are true or false statement. So we will repeat the same thing that it can be said that 60% off the class in trouble for the Aborigine woman. All 2000 scores. Isn't that angel off 3375. This is the range toe 4000 and 25. This is given to us already. So therefore, women say that it may be correct that 60% off. So I'm concluding now that 60% off the 3000 institutions also off higher learning in the U. S. On gold between, say, 3700 plus or minus 3 25 students. So that means the therefore say that this statement is also true. Then we wanted the next part. That's the last part for this question. But e we're supposed to check the normal cough can be used to figure confidence levels. Here are the door because the data don't for the normal cope. So Teoh, right? My answer on this part I'm saying that the statement that the normal cough can be used to figure out conference levels as the later for a normal coach here and well, you see that the statement is falls. So the reason you what is that? Uh, when the sample size increases. So in this case, the detail approximates toe normal distribution. Norman disc um, Yushin. So since I would see the sample spice eyes over here is 400 so which is, uh, large enough to follow normal distribution. That is where we control that. The statement. What they've given in the question is for us.