Problem

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02:37

Problem 55 Easy Difficulty

$$
\begin{array}{l}{\text { Which would be hardest for you to give up: your computer or your television? In a recent }} \\ {\text { survey of } 1677 \text { U.S. Internet users, } 74 \% \text { of the young tech elite (average age of } 22 ) \text { say }} \\ {\text { their computer would be very hard to give up }(P C \text { Magazine, February } 3,2004) . \text { Only } 48 \%} \\ {\text { say their television would be very hard to give up. }}\end{array}
$$
$$
\begin{array}{l}{\text { a. Develop a } 95 \% \text { confidence interval for the proportion of the young tech elite that }} \\ {\text { would find it very hard to give up their computer. }}\end{array}
$$
$$
\begin{array}{l}{\text { b. Develop a } 99 \% \text { confidence interval for the proportion of the young tech elite that }} \\ {\text { would find it very hard to give up their television. }} \\ {\text { c. In which case, part (a) or part (b), is the margin of error larger? Explain why. }}\end{array}
$$

Answer

a. 0.7190 to 0.7610
b. 0.4486 to 0.5114
c. Margin of error higher for part (b)

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Chapter 8

Interval Estimation

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Video Transcript

this question is based on confidence intervals. What is going to be the formula To find the confidence interval for population proportion? It is going to be PBA plus minus Z Alfa by two into route over fever into one minus fever fever into one minus fever upon and okay, what is speed by P bar is my sample proportion where P bar is my sample proportions. My sample proportion. What is N n happens to be my sample size, my sample states. Okay, so let us go to the questions. We want to develop a 95% confidence in double. All right, Okay. Moving on to part A. This is a 95% confidence interval, which means my Alfa is going to be 0.5 And my Alfa, by to my Alfa by two, is going to be 0.2 Faith if I use a critical value calculator, what is the value for Z Alfa by two? That I get the value for Z. Alpha beta is 1.96 Okay, now what is the proportion that is given to me? Okay. In a recent survey off 1677 us Internet users, 74% of their young tech elite say that their computer would be very hard to give up right, and only 48% say that the television will be very hard to give up. So we want a 95% confidence interval for proportion of young tech elite that find it very hard to give up their computers. Between P bar is 950.74 my P bar is 0.74 So if I directly put this into the formula for confidence interval, this is going to be 0.7 food plus minus Z alphabet with 1.96 into Rudo er 0.74 multiplied by one minus fever, which is 0.26 upon route. And what was the value of n 16? 77 right? This was my value off end, right? Yes, 16, 77. So if I use a calculator, this is going to give me a parent lower limits as 770.719 point 610 So my endeavor will be 0.719 to zero point 761 761 This is where 95% confidence a double in part eight Part B moving on part P. I want to develop a 99% confidence interval for the proportion of young chocolate that find it very hard to give up their television. Okay, so this is 99% confidence interval 99% confidence interval. This means my Alfa is 0.1 0.1 and Alfa by two is zero point Zito, Zito fight So if I can put it the critical value it is 2.5. This is 2.5758 Okay, what is my PBA? My people in this case is 48%. So let me write this point for it. I went to simply put it in the formula for confidence. Interval 0.48 plus minus Z Alfa by two is two point 5758 multiplied by route over. This is 0.48 multiplied by 0.5 do upon route starting upon end. Right? We know all of these values. What was in in was 1677 1677 All right, so if I use a calculator for this, this is giving me 0.44862 point 511 for my conference and travel is 0.4486 2.511 Food. All right, Now that's more the party. In which case is the margin of error larger. Well, the margin of error will be definitely larger in case of 95%. Right, Because 95% means what? That this total area in the tales is around 5% right? 0.5 with me that this area is larger, right? If this area is larger, it would mean that the area in the middle is smaller. On this area is 0.95 in case off 99%. This area in the middle would be 0.99 Okay, so where will the margin of error be? Larger. Margin of error. Very well. Very well. The area in these tales be larger in case of 95%. Right? So this is a party. So where is it? Larger. It is larger in party. Why is this so? The reasoning in case of 95%. So this would be my answer

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