Q2 Calculate the necessary sample size that will be used t0 estimate true average porosity t0 within 0.2 with 99% confidence interval if the standard deviation is 0...

Find $n$ for a $90 \%$ confidence interval for $p$ with $E=0.02$ using an estimate of $p=0.25$.

Yeah, we'll be using the central limit theorem and the empirical rule for this problem. So we know our sample is going to be normally distributed with the mean at the population mean, and we want to be to have 99% probability that are Sample average is within one c. So 99% for the empirical rule is three standard deviations above and below the me. So we want three standard deviations below to be our population average minus one, and three standard deviations above to be Population average Plus one. So that means each one of these standard deviation markers are a third. So that tells me my standard deviation is 1/3. So I'm going to plug that in here, one third is my standard deviation. We know our population standard deviation is 10, and we're looking for the sample size to go along with that. So to solve this equation, I'm just gonna cross multiply this real quick one times the square root event is the squared event, and 10 times three is 30. And then to get rid of the square root, I'm gonna square both sides so that our sample sides comes out to be 900.

All right. And this question we are looking at finding the minimum sample size needed for um achieving a given margin of error. And on this one we're using the same formula as before. So first thing we have to do is figure out our alpha, which if it's a 95% confidence, that's gonna be a 5% alpha. We split that in half to figure out our Z score based off of the appendix table in the back. So I'm gonna go to tea with a tail probability of .025. I'm going to scroll all the way down and I get 1.96 which is always the Z score for a 95% confidence interval, which is the most common one used. So then I'm going to plug in to my formula 1.96 squared times 0.5 Tom's .5 divided by my given margin of error squared. And we can enter that into our calculator and copy and paste will be your friend on these And we get 2401 for part B. Were changing too. A Um 99% confidence in it. Well, it's the only thing changing. So going back to our table or tail probability is half of a percent um which is to .576 2.576 Squared times 0.5 thomas 0.5. Remember we're using our most conservative estimate for P. Hat, which is .5 because it's not given otherwise. And then our margin of error I believe is still oh two Squared. Yes it is. And we can get our sample size clearly goes up rather significantly. We always round up one because we're talking about people here, we round down our margin of error is going to shift down a little bit and we don't want that. And then part C. again a 95% confidence, So 5% alpha, which is the most standard we use, Um which is the 1.96 plugging into our formula and our margin of error is smaller, so it is going to go down and we can copy and paste that into our calculator, like what we had on um part A and just change the point to To be much smaller at .1 and we get 9604 would be are needed sample size for that one.

So for this question will be using formula who's he squared times P. That modifying one minus P. And divide all that by E squared. So I started with per a. The P value is .81. It is a 95% confidence interval And the e. value is .02. So before we put all the numbers of formula we need to get Z. And we need to see by changing the confidence in our whole Into a Z score. So first you would change to destinations attracted from one and then get the difference to buy the difference way too. And then that's the number you look up on the sea table and you find out it's 1.96 around it. Yes. Now we can put it in the formula so will be 1.96 squared Times .81 times one minus .1. Then you would divide that all by .02 And you would find out that answer is 1004 out of 78 points six. But because that's like You can have one fraction of this person and a sample size you need to round it up to the nearest whole number and you get 1000 479 for this answer. Yeah. Yeah part beef we use the same formula. So p value it's .81 Lives up to 99% confidence interval. E. is .02. So first again we'll need to change considerable into a Z score. Going to find it carried it .005. And then when you look it up to see table is 2.5 gates. I would put it in our formula. Uh huh. You just do the same um interesting process. There's a switch a number And you get 2,561.5. Again we have to round it up to 2562. Yeah. Then for C. r. standard deviation is eight. Very sorry, P value, P value. It's 21. We have a 95% confidence interval. E. Is too. And because we did this in part a for the 95% confidence in the book, It would be the same number which was 1.96. So then it would be 1.96 squared Tires, Tires, 1 -11 Divided by point there one sq and unit 5012.22. He had to round it up for the minimum it would be 5009 hired 8:13 Yeah.

The following is a solution to number 10. And this says that 210 people were randomly selected. And it doesn't actually say what the population there selecting it from. But it doesn't matter because that sample size of 210 is plenty big, the magic number. There is 30. So I had this been a sample size of, I don't know maybe like 21, then we would actually have to state that it comes from a normal population. But since that sample size is so large, 210, We don't need to say that it doesn't matter how the population is distributed. So n equals 210. It says the sample mean of that population, I'm sorry, of that sample is 20.1 and it has a sample standard deviation of 3.2, and we're asked to find the 90% confidence interval. So first off, we need to determine which procedures should we use and we're gonna use the tea interval. And the reason why we're gonna use the tea interval is because we don't know what sigma is, we don't know that population standard deviation sigma, we only know the sample standard deviation. And uh I know I'm using either the Z or the T. In the beginning because I'm estimating the population, mean it says predict or estimate the population means so it's either going to be the z. Or the tea interval. And the reason why it can't be the Z. Interval is because I don't know what that sigma is. So we use the tea interval procedure and you can either use the formula or any sort of technology you want. I'm gonna use the T. I. T. For because it works out pretty nice. So if you go to stat on the T. I. T. Four and uh over two tests, it's going to be this eighth option here, The t interval, so click eight and summary stats needs to be highlighted there and then we can just start filling in our stuff. So X bar was 20.1. The standard deviation was 32, and the sample size was 210, And we want to be 90% confidence of .9 is our confidence level, and then whenever we calculate this top band here, that gives us our r R interval. Okay, so let's go and write that down. So 19, so this is a pretty, pretty thin confidence interval because the samples has so large and the standard deviation is so small. 20.465. Okay, so that's our answer, so 17 point um I'm sorry, 19.735 and 20.465. Um so if it doesn't say to do this, but if you wanted to interpret this, you would just say we are 90% confident that the true population mean is between 19.735 and 20.465.