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3 - A researcher believes that reading habit of newspaper Is decreasing every year. In order to see the true population proportion of people who are reading newspap...

Question

3 - A researcher believes that reading habit of newspaper Is decreasing every year. In order to see the true population proportion of people who are reading newspaper s/he collect a randomly selected people on the yearly basis. The following table gives the number of people who do not read newspaper with respect to the years Construct a 99 % confidence interval for the true population proportion of people who are reading newspaper in 2010.[20101200NNumber of people who do not read newspaper/90o0

3 - A researcher believes that reading habit of newspaper Is decreasing every year. In order to see the true population proportion of people who are reading newspaper s/he collect a randomly selected people on the yearly basis. The following table gives the number of people who do not read newspaper with respect to the years Construct a 99 % confidence interval for the true population proportion of people who are reading newspaper in 2010. [2010 1200 NNumber of people who do not read newspaper/90o 0.714 , 0.786) (0.173 , 0.227) There is no correct answer (0.214 , 0.286) (0.773 , 0.827)



Answers

Newspaper Reading Times A survey taken several years ago found that the average time a person spent reading the local daily newspaper was 10.8 minutes. The standard deviation of the population was 3 minutes. To see whether the average time had changed since the newspaper's format was revised, the newspaper editor surveyed 36 individuals. The average time that these 36 randomly selected people spent reading the paper was 12.2 minutes. At $$\alpha=0.02,$$ is there a change in the average time an individual spends reading the newspaper? Find the $$98 \%$$ confidence interval of the mean. Do the results agree? Explain.

Right. So to answer this question, we first have to remember how we compute standard error. So hopefully remember to compute the standard era, we first take the standard deviation for one individual in the population which only the square root of P turns one minus p. Then we times by the square root of end to scale up to the population. And then we divide by the population size to turn into a percentage. So if we put this in in all cases, but the question we get the square root event, this is square to 3500. This someone deviation here will just be the square. It's of no 0.71 times, no 0.29 and we divide by the population size we get divide by 3500. And this is exactly what we've done in the question. If you talk this into your calculator, you find that as a percentage, this is that clean no 0.8% or equivalently, nor 0.8 of 1%. So they've done this correctly in the question then else is to find out about the interval 71% plus or minus 1.6 within. So what you should notice is that 1.6% is two times no 20.8% which was the standard estimate. Sorry, standard error that we estimated. And we should know hopefully that if we within two standard errors of our estimate, this goes and 95% confidence interval. So this means that the interval given is a 95% confidence confidence interval for our estimate.

Problem. 13. We have a certain town with a population off 100 thousands, the population is 100,000, and this city produces three different newspapers the newspaper number one and number two and the news paper number three. In a survey, we have the bar abortions for each newspaper and the people who read it. For example, the portion that treats paper one equals 10% off the population. For we were to it equals 30%. And for paper three, it equals 5%. Also, we have the intersections between each newspaper. For example, we have proportion who reads newspaper one and two, which equals 8% for two and three equals 4% and people read one on three equals 2%. And finally we have the portion who reads the three. Newspaper 12 and three equals only 1% for body. We want to determine the number of people who read only one newspaper. To answer this port, we can use Venda Graham when the graham between three different types of papers off newspapers. The first circuit is for one. The second circuit is for two, and third circuit is for newspaper number three to think off this problem. We want to calculate this area. You want to get this area, which means the number of people who read newspaper to only and this for who read only three. And for this who read only one which you can be calculated as we can get the number of people who read the newspapers, all this area and we can subtract from it. We can subtract from it. The area in between here and here and here I am the area between the three newspapers. Let's do this To get this sport, we can use the inclusion, exclusion, identity. We can get it as p off one plus be off to plus B or three. Then we subtract all the intersections between each Ebert minus B off two and three minus P or form and three. Then we add the intersection between all of them. This is for the first board. This is for this port and we want now toe. Subtract these areas. This area's is about it. About the intersection between each beer. We can imagine it as we have this one. Then this one and then this one. You want Oh, to remove all these. But we can see that this area will be excluded three times. And we want to exclude it only one time. Which means we will subtract minus B off one on two. Then be two and three. We can make this two and three and be off one and three. And we will subtract that this area two times because already added three times in two multiplied by P off one toe and three. By doing so, the first term equals 32% minus. The second term equals 10%. Then the total is 20% and toe to calculate the number of population, then population equals or the number of population 20% multiplied by 100 thousands, which equals 20 thousands people. This is for body and for the other birds. We will think for the sports using the same concept for both me. We want toe. See how many people read at least two newspapers for the vendor Graham, At least two newspapers. The two newspapers is here. This area and this area and this area. We want to get this area. If you noticed that in Bardi, we've got this area for the second term. This is required. We've got it as 10%. Sorry, it Z not 10% here It's a 12. 12. Why? It's a 12 Because it equals here It's it four two And here is two multiplied by on gives you two Then eight plus four equals 12 mine plus two minus two equals 12 Then this is 12. 30 to minus 12 equals 20%. This is true and this is okay. We've got this answer from birth. Mhm it equals eight plus four plus two minus to multiply by one which he gives 12 percent. Then the number of people for bar to be read at least two newspaper equals 12% were deployed by 100 thousands equals 12,000 people for Boxee If one on the three are morning papers, we will consider now one and three is morning papers And to is everything paper. We want to determine how many people read at least one morning. Baber Plus on evening, Babe, Let's think using the vendor Graham, at least one morning, Bieber, which means we need this area. This is at least for one. And this, at least for one. And we want the intersection between them, which is this area who read at least one newspaper or two morning papers, plus an evening Baber, which means we will include the intersection between one and two and two and three, which means we will add this area. Let's calculated. But before we calculated, we must canceled the area, which not include newspaper number two. This is the area that doesn't include Baber number two, which means now we have only this area. By calculating it, we can use the intersection between one and two between one and two, and we can use the intersection between two and three. But we have included the intersection between one and two or 32 times. Then we will subtract it one time because it's replicated. It's a topic it equals. We have it one and two. It's eight, and to this is two and three two and three equals well, four. Then we subtract one, which equals 12 minus 1 11 person, meaning that the number off people equals 11 by 100. But the blue eyed boy 100 thousands equals 11,000 people for both D. How many people don't read any newspapers which means for the complete vendor Graham, we won't this area. This area is represented by the total number off people minus the people who read newspapers. The people who read newspaper has been calculated in Bart A in this tour. Look at this. We have calculated the people who read newspapers. It's 32% which means the number of people Hu gone. Three newspapers will equal the total population minus the people who read newspaper, which is 32% off the total privilege which equals 68 thousands of people. And finally, for Bharti, we want to calculate the number of people who root, who read only one morning paper on one evening paper for the vin diagram we have morning paper is one on the three one only one morning paper on one evening paper to read only one morning waiver, which means we read only one or three or three, not both of them and one evening paper. We will include two, which means we want to the intersection between one and two this intersection and the intersection between two and three. And we will not include the intersection between one and two and three because it will make We have make the number of morning Weber's, too. We want only one morning member 143 and the intersection between two. Then it equals. We can get it as the intersection between one and two, plus the intersection between two and three. But do that in both terms. We have included this area and we don't want it. Then we will subtract 22 times 12 and three equals it. The last four minus two multiply it by one equals 10%. Then the number of people for bar E equals 10% deployed by 100 thousands. Then it equals then thousands of people. And this is the final answer off our problem.

In this problem we have three newspapers. Mm The Earth. See you're given that N. E. Is point for N. B. As born 51 N. C. Is 0.68 And the intersection B. Just reading in B. S. 0.30 and be intersection C. Which is reading both BNC is 0.28 and E. N. C. S. 36%. The intersections yes weren't 36 and 8% do not read any of the three newspapers. It means that. And a union be Union see not equal store point zero. Which means that in a union be union C. S 0.9. Now we need the number of people who read all the pre newspapers. We know that and a union be union see is nothing. But in a let's end me that's N. C minus A. An intersection lee witness and a intersection C minus and be intersection C plus and intersection B intersections substitute the values. This is 0.92 equals two. Any person represents C. Is 0.42 Just 0.51 Yes, I said. Which is 1.61 minus. This is 130 point to it 1236 US and a intersection section C. And in the section B intersection C. Yes, nothing but 0.92 That's 0.30 This point to it right 86 which is 28 6 minus 1.86 minus 1.61 which is born 25 which means that people who read all the three newspapers That is the intersection is 25%. That's all.

Here we're going to look at how many we need in a sample when we want to estimate the population mean and the margin of error is a fixed margin of error requested of us. Now we're also going to then compare some different situations that are involved in this so we can make some observations about what happens when our margin of error is um it has changed. So this says how many individuals are needed in the sample to estimate the population mean. Number of books within four books with a 95% confidence interval for part a for part B. Within two books With a 95% confidence interval if s is equal to 16.6 when we are estimating the population mean and the error margin of air is given to us. The formula we use is N is equal to Z sub alpha over two times S divided by e quantity squared. And we need to make sure that we round up to the next hole number after we calculate this so that we meet the requirements and that we answer it in uh how many we need in our sample type of an answer. Now, this involves the critical values from Z. So from the normal distribution and the way you find those critical values was covered in another video but just as a reference so that you have this information handy. It's good to have down that the Z sub alpha over to critical value For a 90% confidence interval Is the critical value of 1.645. Yeah, the Z sub alpha over to for a 95% confidence is 1.96. And the Z sub alpha over to for a 99% confidence Is 2.575. Now for part A. We are looking at finding the number in our sample And it's 95% confidence. And when you're reading these types of problems, what follows the word within that is what is your margin of error? So when it says within four books are margin of error. E. is for in this part egg. So our n. is equal to the quantity 95%. It's 1.96 Times S which is 16.6 divided by Ian Part A. is four quantity squared. And when you go and calculate that through You get 60 66.16 but you have to round up to the next hole hole number not round nearest round up. So this end is 67 for part B. We're doing two books Within two books so it's still 95%. So it's still the 1.96 Times The 16.6. But my margin of error that's required of me this time is to one of these squared and when we go through and calculate that we get N is equal to 2 64.65. Rounding up we report and is equal to 265. Now what happens when we doubled the require accuracy? So I had an accuracy of four um books and then I had an accuracy. That is my two books. So actually we are looking at that if we doubled our required accuracy it actually quadruples the sample size. What now? Another question is what if we were looking at within four books? But it is a 99% confidence level instead of a 95%. Well if that's true then it's the critical value that changes. And for 99% the critical value is 2.575. So our n. is going to be 2.575 times the 16.6 divide by my four. Remember what follows the word within. That's your e quantity squared And we get 114.19. But again round up to the next hole number and is equal to 115. And when I compare that to a. Because remember that's the four books, It was 67 when I was doing a 90 Um 5% confidence in her vote with four books and it's 115 when I'm doing a 99% confidence with uh four books. So my sample size increases as the level of confidence is increased. Yeah. Right. Mhm. Oh


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