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A researcher would like to estimate the proportion of home-based businesses owned by women. What is the minimum sample size needed to be 80% confident with a 3.5% m...

Question

A researcher would like to estimate the proportion of home-based businesses owned by women. What is the minimum sample size needed to be 80% confident with a 3.5% margin of error? (Round your answer to the appropriate integer.)Answer: n =

A researcher would like to estimate the proportion of home-based businesses owned by women. What is the minimum sample size needed to be 80% confident with a 3.5% margin of error? (Round your answer to the appropriate integer.)Answer: n =



Answers

In a survey, the planning value for the population proportion is $p^{*}=.35 .$ How large a sample should be taken to provide a 95$\%$ confidence interval with a margin of error of .05$?$

Given C, he couldn't. 95% equal 0.95 e equal 0.1 People are equal 0.2 five form the size formless sample size D forward None and equally the all for over to but power to mhm multiply, boy B forward. Thank you both. Over the square. Equal. Mhm. Okay, Did all for over two old power to multiply, Boy P forward multiply, boy, Want minus B forward over Mhm. Okay, E square, equal the all four over tool. Cool. Pour too multiply. Boy 0.25 Overeat square. What for? Confidence 11 1 minus all for equal 10.95 Did your mind? Is it out of over two equal, Is it 0.225 Using paper five. Look up 0.25 in the table. The that school is in the found. That school with opposite time through is it all for over two equal 1.96 people is known. Then the sample size is round up in equally Read all over to cold War to multiply, Boy P power multiply by one minus p power Mhm e squared equal. Yeah. One. We went 96 or two. Boy 0.25 boy one minus 10.25 over 1.1 square. Almost equal 73. So in equal 17. 3. Question number B Mhm. Uh huh. All right. Given C equal, 95% equal 0.95 e equals 101 people equal 0.25 We will use a formula size, um, of P. So for confidence, level one minus alpha equals 10.95 It reminds it also over two equals it, um that 0.25 using table five. Look up 50.25 in the table. Does that score is in the found? That school with opposite time. So is it all for over two? Equal? 1.96 people is unknown. Then the sample size is ground up and equal Did all over to old people too. Multiply, boy boy 25 over e squared substitute 0.1 for E. So this will equal and 9 97 Mhm

The following is a solution in # five. And uh this asks a sample of 300 people if they work at home at least once a week and 35 of the 300 say that they do. So the number of favorable is 35. That'll be our X. And then the number of total would be 300. So if you wanted to find the p hat you can it's just 35 over 300. And we're asked to find the 99% confidence interval here. Now this one's a one proportions Z. Test so it's one prop Z. It's not test. I'm sorry interval. So one prop c. ent is this case because it's a proportion it's a favorable over total. We're looking at the percentage of people that work at home at least once once a week. Okay. So we're gonna do the 99% confidence and you can do this with the formula that may take you a little bit longer. I'm gonna use the T. I. 84 because it works pretty nicely. If you go to stat and then test and then go all the way down to the a option where it says one prop. Z ent All right. We got one proportion and we're just making the Z. And And this is where we get the x in the end. So the x was 35 And then the end was 300 And we want to be 99% confidence. We'll say 9 9. And whenever we calculate that's going to give us this top band here, that's the that's the confidence interval. And if you wanted you can have the P hat. That's about about 12% is the P hat. So let's go and write, write down that that interval there. So I'll go in around generally speaking I like to around 11 day, so 6.9% or .069. So I'll go to three decimal places here. Um so 0.692 point 164 with the numbers. So if you wanted to interpret that, we didn't have to because they didn't say, but if you wanted to interpret this, He would just say we we are 99% confident that the true proportion of people that work from home is between 6.9 and 16.4%.

All right. So we're giving a sampling. Were given a sampling distribution where our sample size is 40. Our sample mean is 25 under standard deviation for the population is five. So for part A, we want to find our standard error for the mean and because and is greater than 30 by central limit their, um you know this distribution to be approximately normal, and this is key, because then we can use our formula for standard error, the mean being population standard deviation divided by the square root of the sample size. That's gonna be five divided by square root of 40. And if you do that out, that's gonna be 0.7906 All right. Part B were given a 95% confidence in a confidence level. We're not finding an interval on getting ahead of myself, and we want to find the margin of error e So a 95% confidence level give us an Alfa level of one minus 0.95 or 95% which is 0.5 So from our formula, we need to know that the Z of Alfa over too times standard error for the mean. We look at our ze table, which I just so happen to have on the right hand side of this screen. I'm gonna be looking for Alfa over too. Is 0.0 to 5. So what I'm really looking for is 0.9750 Because if we dry out, this normal graph is gonna look something like this. It's supposed to have 95% of our values. So that means that we should be looking for because that's gonna be 2.5% here and 2.5% here We're really looking for in 97.5%. So where's that undersea table? Well, that's gonna be 1.960 right there. Which is good fact to remember in general, because 95% is a fairly common confidence interval. So anyway, that's gonna be 1.960 times 0.7906 And when you do that out, that's gonna be 1.549 It's five, I believe There you have it

One of the formula for margin of air and E is equal to two time to square root P times one minus p provided by then. So if we have peace, you got 72% also known as 0.72 and and it's equal to 100 we can plug p n n In to this equation, we have to come to square root. 0.72 times one, Linus 10.72 divided by 100 is equal to two time to square room of 0.72 times one minus 10.72 which is equal two 0.202 divided by 100. And if we put this into our calculators well again, it's approximately 0.9 or nine percent.


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