Hi. Here. In this question we have Bean asked you to find the confidence interval estimate for population standard division and two types of data is given accordingly. We have another question to find which system is better. Okay, So first part of the customers start with the first part. The first part. We have been given information, but 6.56 point seven, 6.6. 46.66 point 76.8, 7.1, 7.3, 7.4, 7.77 point seven aunt against seven points around. So these are the observation is given. These observations sort of present the customer under single waiting like Okay, so question is asking us to find 95 posts in this confidence in gravel. Estimate for population standard division. Okay, so here we have to obtain the sample. Mean sampled, off sample, standard deviation and everything. Then we have to obtain the chi square value. Then you have to find this standard deviation estimate. Okay, So, first to be find the mean value that this expert expert you know, we can find by using Sigma examining it. Okay, so here you can see my excitements. You have to add all these values 6.5 plus 6.6 plus 6.7 plus six dropped to 7.7 the last year. You have to add by number observation. You can see six plus 4 10 observations after So when you find extra, when you add everything you get 71.5 better bite, so you obtain the mean values equal do 7.15 so this will be done. Required Mean value for the observation. In the second step, we find the standard deviation. So to obtain standard deviation standard deviation Yes, we know the formula is given by Sigma X, a minus expert. The whole square right and minus one is the formula for finding standard division off the given they doesn't so first observation. Six point files will be finding 6.5 minus 7.15 square plus Second observation is 6.6, So we take 6.6, minus 7.15 square plus x. A problem. The last term of 7.7 minus 7.15 square will be adding the difference off squares off, observation minus. It's been by end minus one. And you know it is 10 here, so and minus one musical with 10 minus one. So we get this value. After simplification, we get the values equal toe 0.4767 This will be required value? Yes. Once you get these two values in the next step, we are to obtain the critical values tow, obtain critical values. We're going to obtain Chi Square values here. Okay, so we know end is equal to 10, then degrees of freedom of vehicle do and minus one that is equal to nine on confidence in probably out to make 95% conferences. Right. So, Alfa Valley, we can say it is equal to one minus C that is equal to one minus or 95. You get a 0.0 fight, find 05 So the first guy square bag that is left, Dave that child critical value given by Chi Square one minus Alfa by. But you can write Chi square one minus alphabet. You can write us one minus 0.5 by two. You get 0.0. What is your point? 975 You obtained this value is equal to 2.700 from the chi square table. Next to be find a right killed critical value. Right? Tell the critical value you can obtain Chi Square Alfa by two that is equal to Chi square 0.25 from the table we obtained that vary also 9.4 90.0 to 3 Once we get critical values in the last year in the first data set, you can find these sigma values you right So the formula Now we're finding confidence in trouble so to find confidence in trouble for standard deviation the formula is given by sky rodeos and my restaurant by Chi Square Alfa by two times Standard division here we have and minus one by Chi Square one minus Alfa by two times Standard division Okay, so we just have to substitute these values. We get the square root off and minus one by I square value. You know it is 19.0 to 3. Thanks. Standard deviations 0.47 67 less than standard deviation less than squared off and minus one by second value Obtained 2.700 right time standard deviation 0.4767 So after simplification, we get the value for estimate off. Standard deviation given by 0.33 Here, we can still zero point 87 week. Um, so this will be the and support first part of the question. So we're getting the estimate 0.33 to 0.87 Now, The second part of the question Second part of the question, we have been given the data. Okay, we'll take the data. Nephew second. Yes. So here we have been given the observation for this Second bank is given 4.25 point four, 5.8, 6.26 point seven, 7.77 point seven, 8.59 point three on Finally So, as usual, the same procedure will be following possibly find expert, then standard deviation and case core values. And finally, the confidence interval. So now we know the procedure so expert will be given by Sigma excited by. And so we get the value. We obtain all these values directly. No. Okay, so now beginning 4.2 plus 5.4 plus etcetera plus 10. You add number of terms again here also tend only So we get the mean values he could do. 71.5 is coming here by 10. So 7.15 will come. So here also me in the same Or you can see that means waas Here are the 7.15 the second case also get getting the same meeting. Okay, The next step will be finding standard deviation value So standard deviation You confined by using Sigma X I minus expert the whole square bite and minus one right square root system That is it will do so 4.2 is there minus you. Take 7.15 square second Observation ISS 5.45 point four minus 7.15 square, etcetera. The last observation is 10 10 minus 7.1 Favor square, take a square root you want to buy and minusma. Okay, so we get this observation value Standard division value. We get 1.8216 Welcome. Okay, So next step will be finding high square values. Right? So to obtain high school values we know anything could turn here so degrees of freedom will be and minus one that is equal to nine again. The conference and double be required US 0.95%. This Oregon, right 0.95 Here. Right now the left kid Critical value will be given by Guy Square one, minus alphabet. We have to find Alfa here. Alfa is what? One minus 0.95 that is equal to 0.5 So here we get Chi Square off 0.51 minus you take means you'll be getting 0.9555 again. Right? So the value obtained 2.700 Similarly, you can find right tailed guns particular value. That is, until do Chi square Alfa by two If he could do Chi Square Alfa by two we get 0.25 year again on the value obtained is 19.0 to 3. Good. No, I'm Finally We're going to find the confidence interval estimate the conference in level estimated for standard deviation. For that we know the formalized and minus one Bye bye Square Alfa by two times standard deviation is here we get and minus one by Chi Square one minus Alfa by two times standard division. So just substitute all these values we know already. And my investment values. We know it is nine here by ice car value obtained 19.0 to 3 times Standard division. You know it is one point 8 to 16 a big Mac again and minus one and minus one is 10 minus one bite 22.7 times standard deviation. We got 1.8216 So when you simplify this value, we get 1.5 sigma three point greatly. This will be the standard deviation estimated for second I'd given. Okay, No. Once we found the answer. Four questions, second part No, in the park. See? It is asking us toe interpret this result founding party at part B. Right? So the closeness, whether this confidence in trouble suggestion difference in the variation among the waiting times in that banks Andi Challenge mint is better. Is asking whether a single line system is better or multiple line system is better. So you can see, while calculating the mean values we got both the means. Values are safe here. We got 7.15 in the first case Also, we got mean values 7.15 So both means our state. Right? So both means the same means we can see what the confidence in the way part off a which is does not overlap. Confidence in trouble. In part era does not overlap that both. But be okay, because the result variation among the waiting times here. Right then next we can say so that there is variously in waiting times. So the single system between system will be better is asking. We can say single line system will be better. What is the reason? The reason is here the system both systems are having saved me. So it doesn't matter. You're having a multiple language because same Mean is the right. So both system is having same mean so single line system will be better in this case. Okay, this will be the answer. I hope this answer their question. Thank you.