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Researchers working in a remote area of Africa feel that 48% offamilies in the area are without adequate drinking water eitherthrough contamination or unavailabilit...

Question

Researchers working in a remote area of Africa feel that 48% offamilies in the area are without adequate drinking water eitherthrough contamination or unavailability. what sample size will benecessary to estimate the percentage without adequate water towithin 6% with 95% confidence

Researchers working in a remote area of Africa feel that 48% of families in the area are without adequate drinking water either through contamination or unavailability. what sample size will be necessary to estimate the percentage without adequate water to within 6% with 95% confidence



Answers

Overseas Travel A researcher wishes to be 95\% confident that her estimate of the true proportion of individuals who travel overseas is within $4 \%$ of the true proportion. Find the sample necessary if, in a prior study, a sample of 200 people showed that 40 traveled overseas last year. If no estimate of the sample proportion is available, how large should the sample be?

I want to be 95% confident. I want to be 95% confident that my sample mean is within one unit of population Mean that my sample mean is within one unit, which means that my e is equal to one. Right? I will say that my easy for the one I think it has to be equal to or less than one. But let's say that it is equal to one. And my sigma is 48 my sigma is 4.8. So not 48. This is 4.8. Okay, What is the formula for standard editor? He is equal toe z l five by two sigma by routine Now n is the value that I wanted to calculate. Okay, 95% confident means my Alfa is 0.5 My Alfa is 0.5 But I can say my Alfa by two is 0.0 to find right now. Yeah, If I use a critical value calculator, I get my Z values E Alfa by to be 1.96 So I just substitute the values here. One is equal toe 1.96 into 4.8 upon route. And right now, if I use a calculator to calculate this, my en is going to be 1.96 multiplied by 4.8, which is 9.408 And if I square it, I get 88.51 I get any is equal to 88.51 88.51 But since and has to be an in teacher, I will take this as 89. So this is my minimum sample size 89. This is my answer.

All right, This question wants us to estimate the sample size needed for a proportion. Where all it gives us is the margin of error, and the confidence level doesn't tell us anything about P hat. So when that happens, we let P hat equal 0.5 because that maximizes the variability. So it gives us the worst case scenario. And now, from there we'll use the equation and equals. He had times Q hat times the error divided by our special Z score Look that our special Z score divided by a margin of error all squared, which in this case equals 0.5 times 0.5 times 1.96 over 0.3 quantity squared, which in this case, when you multiply everything out, you'll get 10 67.1, which with margin of error, questions of sample sizes. We always round that up because if we round down, our margin of error will not be big enough. So 1006

19 the critical value for Table six, the degrees of freedom equal toe end minus one, which is 19 minus one. So is equal to 18. The chi square one minus all going toe to five, which is the chi square off all 0.975 which is equal to 8.231 And the chi square off 0.25 is equal to 31.5 to 6. So for the boundaries for the Thunder D V engine, which is in minus one over Chi Square well for over two times s, which is 19 minus 1/31 190.5 to 6 times 15, which is equal to 11.334 and end minus one over. Chi squared off one minus all for over two times as she is equal to 22.1 82 So these are the boundaries for the standard deviation. For the variance, it will be the square off these values, which is 11.33 four square and 22.218 to square, which is equal to 1 to 8 0.46 um, and for 2492.41

We want to calculate at 95% confidence the population proportion P with an accuracy within 5%. We want to find the minimum sample size and needed to accomplish this for a no estimate of the proportion P hat be a test statistic of P hat from a survey results or other data of 50.28 And see we want to compare the answers from A and B. So to get started we need to be able to solve A and B. Using the formula if I hear and equals P. Hat you hat times over E squared E. Is the marginal 0.5 M. Z is the critical value related to our confidence level which we can find from a Z. Table or a standard normal distribution. So Z of 95% is 1.96 So now we plug in to a. Where he had equals equals 0.5. Because we have no estimate we obtained and equals 385. For B. We plug in P had equals 3850.28 Q is equal 2.72 and we get and is 303 comparing answers. We see that ended slightly larger for NOPD.


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