Question
(D) What is the point estimate of the difference between the two population mneans; (Use mean score first round mean score fourth round_For which round is the population mean score lower? The mean of the fourth round scores was lower than the mear of the first round scoresThe mean of the first round scores was lower than the mean of the fourth round scores
(D) What is the point estimate of the difference between the two population mneans; (Use mean score first round mean score fourth round_ For which round is the population mean score lower? The mean of the fourth round scores was lower than the mear of the first round scores The mean of the first round scores was lower than the mean of the fourth round scores
Answers
The following data are from a simple random sample.
$\begin{array}{llllll}{5} & {8} & {10} & {7} & {10} & {14}\end{array}$
a. What is the point estimate of the population mean?
b. What is the point estimate of the population standard deviation?
Hello. Everyone in this video we're going to identify that statistic does used for population mean and standard deviation for mean. The notation looks like a you that has a L. On the left side, that's the notation for population mean. And the formula is the Some nation think of some nation as adding all the numbers up together. X I X I represents each number individually that we have. And then some nation is just adding all the numbers up, dividing by end and represents how many numbers there are. Or you can count based on positions next for population standard population standard deviation. We have a notation that looks like this kind of like a circle with L. O. Tick that equals to the square root of there's some nation of X. I. Which is each individual observations minus are mean that we found over on the left side of the white board squared so make sure that you're taking the observation minus the mean then squared before doing the some nation. Some nation is the collective um process of you taking each, each and every individual observation minus ng the mean and then scoring it and then adding everything up and then divide by end, the number of observations. Don't forget the very last step after you divide to some nation by the number of observations you have. Make sure to find the square root of that number.
The first thing we have to compute is the differences between each element from each population. So 11 minus eight is equal to three seven minus eight is equal to negative. 19 minus six is equal to three. 12 minus seven is equal to five. 13 minus 10 is equal to 3 15 minus 15 is equal to 0 15 minus 14 is equal to one. And now we have to find this is the answer to part A. Now we're asked to find the bar which is just the mean of the differences so that is equal to each individual mean the sum of each individual need over the number. Sorry. The sum of each individual difference it over the number of differences which is equal to In this case, the sum of each of these individual data points over the number elements which is seven, which is equal to two. And now we have to find a sample standard deviation which is equal to the square root of the some of the difference between each individual difference and our mean difference squared over the number of differences minus one soldier to a new page. For this equal to the square root of the Somme. Each individual difference minus mean difference over a number of differences minus one which is equal to approximately 2.817 This is the answer to part C. And now we're asked to find a point estimate for the difference of population means. And this is so the difference of population means is also our deep are which is equal 22 And now we have to come up with a a confidence interval. So we're asked to find a 95% confidence interval. So to do this, we will use, um, the following formula. Our confidence interval equals D bar plus or minus a T to t statistic. Because we're not given a population ah, standard deviation. We compute the sample standard deviation a T statistic for Alfa over to where Alfa equals one minus the confidence level. So that is equal to one minus 10.95 equal 2.5 So de bar plus or minus our T statistic for half of our Alfa times wth east andr deviation, the difference is over our sample size. So in our situation, D bar is equal to two plus or minus. I'm just going to write t of 0.0 to 5 for now because Alfa over to is 0.5 over to which is your 0.0 to 5 times 2.817 over the square root of seven. In order to find our T statistic, we have to compute a degrees of freedom and our degrees of freedom is equal to end minus one, which is equal to seven minus one, which is equal to six. Now, using a tea table, we can find out where a significant Slobo of Sierra 0.25 associated with a degrees of freedom of six lies. And we get that we have a confidence interval of two plus or minus 2.447 times 2.817 over the square root of seven. And that means our final answer is our were 95% confident that the true average difference for our two populations lies between our lower end of our confidence. Interval 0.747 and our upper end of our conference interval 3.9253
YEAH hair hair For the solution Let x one x two x three x four x five and x six are the discourse. So um, you will be equal to x one plus x two plus x three plus export plus X y plus x six Divided by and here is the numbers. So we know that there are six numbers so mu will be equal to x one plus x two plus X three plus export plus X five plus access divided by six. Now, given that five scores are each above the mean by one point so new which is equal to x one plus x two plus x three plus x four plus X five plus x six divided by six will be called to her. We rearrange this as six new equal to mu plus one plus new plus one plus new plus one plus mu plus one plus new plus one plus x X as the five schools are each above them mean by one point. So now again, simplifying this we get six you equal to five mil plus five plus success So six x six equal to mu minus five. Therefore, the sixth it's called is below the mean by five point. So this is the compared solution it step by step, Please go through this thing.
All right, So we're given this set of eight points that comes from our sorry eight data points that come from a norm population. And we have some questions. Ask. So first off, what's your point estimate for the population mean or a point estimate for the population mean sorry. Equal sign is not technically after it. It's gonna be whatever sample mean is, uh, when you have these up, this becomes 10. 18 30 45. 50. There should be an 11 in here. My bad. So 10 18 30 45. 50 uh, 56 67 80. Yep. That was just a sandy check for me. Just making sure I had all the plot point that anyway, So All right. What's your point? Estimates. Come on. There we go. For standard deviation. Well, that's just gonna be the standard deviation of the sample, which, if you do the mathematics for which I'm not going to do by hand right in front of you, you're gonna get three points. 464 All right, now to find hour interval at and 95% confidence. So we want to find out margin of error in party and because we do not have a given population standard deviation. We only have ah, point estimate. Oops. We're going to use a tea interval. So it's gonna be tea time. Standard deviation over skirt and equals eight. We look a t star, we have n equals AIDS. So our degrees of freedom, which is an minus one, is gonna be seven. We check our tea table at seven degrees of freedom for 95% confidence. We get, uh, 2.365 All right, so this equals 2.365 times 3.464 divided by the square root of eight. Punch that in on our calculator and we get So this is about 2.8964 All right, finally, we're gonna constructor confidence Interval. All right, so Oops. There we go. So 10. Plus, he's simple. It's just 12.8 964 and subtracting out. You'll get some 0.1036 There's your interval. My pen's acting weird. Come on. There we go. And there you have it