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Sampie Ol 31 obsenvations taken tromn normal populaton with sandard deviation &l 16, The sample mean is 39 Determine Ine 80% conficence interval for the populat...

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Sampie Ol 31 obsenvations taken tromn normal populaton with sandard deviation &l 16, The sample mean is 39 Determine Ine 80% conficence interval for the population mean (Round the final ensuers t0 decimal places |Coniidence Interva iar the populabon mean

sampie Ol 31 obsenvations taken tromn normal populaton with sandard deviation &l 16, The sample mean is 39 Determine Ine 80% conficence interval for the population mean (Round the final ensuers t0 decimal places | Coniidence Interva iar the populabon mean



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Random samples of size 64 are drawn from a population with mean 32 and standard deviation $5 .$ Find the mean and standard deviation of the sample mean.

The given Parameters being a sample of 15 sample means 1 3 pointed samples. An additional 6.3. We have not able to amuse 25 already. But you didn't build only 25. 0.01. So we have given 10,000 division. I mean we have to go 14 days. Right? Students the despair to go go. So this book for that the first parliament that has given us P. That's coming up to be people. Thanks for negative mean. Or we have to deal with. Okay For the biological reality that cannot be negative. 0.73 years Within technology they find that the value for two tailed the only we had a commander with your point 4729 people just coming up to be 0.47. I don't know. And the P. Value check now B. F. Uh We have greater than other parts right? Therefore we spend so or reject and peter Bergen alcohol to repair to reject and North That means meal seems to be 25. So we can say that population need is not a friend from veronica. Population means is not different. 25. Thank you.

So in this question, we're told that a population has a mean of 16 and a standard deviation 1.7. And first we asked for the mean and standard deviation of the sample means, so that's mean of the sample mean is just the same as the population mean. The standard deviation of the sample mean is standard deviation of the population mean over square root of the sample size, which is 1.7 over Route 80, which is .1901. But we were asked for the probability that a mean of a sample of size 80 Will be more than 16.4. So we have the probability that are standard moment when the variable is more than 13.4 -16 over .1901. So That value here 16.4 -16 Over our standard deviation. So that value is 2.1 zero Or 2.11. So that value 2.10 which is basically one minus the probability Z is less than 2.10 So z being less than 2.10 is .98-1, So 1 -1821 is .017, no

For the given parameters, soviet, given years, uh, example being the samples and attribution Publishing Media. We have already businesses that Russian is better than a 100 and european european level of significance and supported the tragic here he's given us expand negative meal or s by. Okay. Oh just so all this These are things coming up with three point obviously three points. No they don't. Now the p value at this probability value at this technology here 0.0 to tweet is right and the value 0.00 country that is the right edge P value zero point with a little boat. Now he's compared me with alpha and has two piece coming out to be less than open. Right. Therefore really Jack now the bulletin so we reject the politics. That means we can say that there is sufficient evidence that coalition meat is better than 100. There is sufficient evidence that population means is greater than a 100 from you.

In Exercise three, we're going to be considering a random sample that is drawn from a normally distributed population and this time the standard deviation is unknown and we've been to Use the given information to construct a 99% confidence interval for the population mean? So we have two cases in both cases, the sample uh mean and the sample standard deviation are 386 and 24 respectively. For the first case the sample size is 18 and the second cases sample size seven, So it's a smaller example for the second case, no first things. The first thing we need to do here is to determine the level of significance And also to determine the right formula that we need to use to get the 99% confidence interval. Now, since the uh population, standard division is unknown, we're going to use the following formula X. Bar plus or minus The critical value of tea for two tailed tests, multiplied by s example, standard deviation divided by the square root of n. And since it's a 99% confidence interval, it means that uh the value Of Alpha, the level of significance equals 1 -0.99 And that equals 0.01 Half of that is 0.005. So this is useful for us to determine the correct critical bottle 40. Now, if you're using the t distribution, you also need to give the degrees of freedom for each case. Now, for the first case, Because the sample size is 18 So that the number of degrees of freedom will be N -1, which equals 18 -1 and that equals 17. So you want to check for the corresponding critical value of T Where they for 0.0 Uh five as a level of significance. So that corresponds to the value two point eight 98. And now that we have all we need we can substitute um all the material and all the content that we had uh into the formula. So let's begin. So X bomb is 386 plus or minus. The critical value 40 is 2.898, Multiplied by S which is 24 Divided by the Square Root of N., which is square root of 18. And when you saw that the calculator you can to obtain 386 plus or minus 16.39. So the margin of error for this case is 16.39. Now let's proceed to the second case, We want to start by getting the degrees of freedom which will be N -1. And in this case it's going to be 7 -1 and that equal six. And so the critical value of T. There corresponds to this level of significant European 005 Is uh is obtained from the table, it is 3.70 uh 7. Then We can substitute our values into the formula again and you're going to get 386 plus or minus 3.707, multiplied by 24, divided by the Square Root of seven. and when you work that out, you're going to get 386 plus or Uh minus 33 0.63. So for the second case where we have a smaller sample, you can notice that the margin of error is greater, meaning that we're going to have a longer interval is compared to when the sample size is smaller.


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