Okay. So in this case, the number of chocolate chip cookies is approximately normally distributed with a mean of 1 to 6 to with men of 1262 Okay, with the mean of 1 to 6 to. And what is the standard deviation? 118 sigma is 118 Yeah. What is the probability that a randomly selected bag contains between 1000 and 1400 contains between 2000 is somewhere around here and 1400 somewhere around here. So what I can do is I know the overall area under the normal curve. When I find the Z statistics for 1000 and 1400 right? The rural area under the normal curve will be one. So what I can do is I can find a P value for 1000, which will be the area and the stale P value for 1400 which will be an area in this stage. And I can subtract both of those areas from one. Right. So this is 1000. This is 1400. So let's just look at this. We are using the Z statistic where X minus mu by sigma gives me the value of C. Okay, this is part of a for X is equal to 1000. What will be the Z statistic for 1000? Italy, 1000 minus 1 to 6 to if I use a calculator. Okay, so this is 1000 minus 1262 divided by 118 which is Sigma. So this is minus 2.2. This is minus 2.2. Now, how do I find a probability for this? The probability for the number of chocolate cookies or a number of the cookies in the bag being less than 1000. It will be given by the P Value for this. How do you find the P value? We can use a set statistic table, A Z table, or and online to a lot of statistical package like Excel. So this is what I'm doing here. I put in the value of 2.22 03 I had enter and my P values 0.13 So this is 0.13 Right. So this area is zero point zero and three. We are going to do the same thing for 1400. So what is the Z statistic for 1400? This is going to be 1400 minus one, 262 and divided by 118 which is 1.1694 This is 1.1694 What is the corresponding P value? 1.1694 Okay, I put in this value. This is 1.1694 they had entered. And what the value is 0.1 to 11 This is 0.1211 So this area is 0.1211 Now, I want the area in between these two. That is this particular area. So I'm going to subtract both of these p values from one. This is going to be one minus 0.13 minus 0.1211 All right, one minus 0.0 and three. Minus zero point 1211 This is 12110.8659 So what is the probability? 0.8659 This is the probability that I want probability that this is between mhm 1000 and 1400. Okay? This is the probability. Let's make this a little neat. Okay? Okay. This is the probability. Now, both of these are inclusive. They say it doesn't really matter much, right? This is the beauty of the set table. Whether they're inclusive or exclusive, the probability will remain the same. Moving on to Part D. What is the probability that it contains fewer than 1000 chocolate chips? Now, we have already calculated the P value for 1000. Right? Which is this 0.13 So what is the probability that are Ex has less than 1000 chips? It is going to be 0.13 Okay, then moving on to part. See? Yeah. Uh, what proportion contains more than 1200 chocolate chips? Okay, so what we want is more than 1200 which means that if I look at the normal curve, this is 1262 right. 1262 This is the mean. We want more than to learn the government. It will be somewhere around here, right? More than means we want this area. So what I can do is I can find the p value for 1200 which will be this lesser area area and the tail, and I can subtract it from one. So let's find the sadistic. For 1200. This will be 1200 minus 1262 divided by 118 which is minus 1180.5 to 54 Negative 0.5254 And what is the P value for this? Okay, negative 0.5 to 54 I hit. Enter. This is 0.2996 point 2996.2996 Now I have to subtract this from one because I want the area on the bigger side. Right on the larger side, this side. So this is going to be one minus 10.2996 or this can be point 704 4.7004 Let's find out 0.7004 actually 0.7004 If so, this is the probability that the number of chips is greater than 1200. Okay, now moving on the part D What proportion of bags contain fewer fewer than 1125 ships contain fewer than 1125 Let's find the Z value for 1125 This is going to be 1125 minus 1262 divided by 118 which is minus 1.16 minus 1.16 Now, what is the corresponding P value? Minus 1.1 six minus 1.16 we had entered. This is 0.1230 Okay, so corresponding P value is 0.1230 What does this say? This is that probability that the number of chips is less than 1125 is 0.1230 Okay, moving on apart. Eat. What is the percentile rank for? 1475 chocolate chips. It was a percentile rank for 1475 chocolate chips. So let's find the Z 24 1475 What is this going to be? 1475 minus 1262 Divided by 118 which is 1.8 1.805 Okay, 1.805 means it is somewhere around here. So what I have to do is I have to find the p value is subtracted from one which will give me this. This is the side of the person time that we want. So what is the P value for this? 1.805 1.8. Do five hit. Enter. So this is 0.355 point 0355 This is the P value, this shaded region, and we also supply this from one. So this is one minus 0.355 So this is 0.9645 8.96 for five. Or I can say that this is 96% time. So this is 96 percentile. Okay, How about back if what is the percentile that contains 1050 chocolate chips? So what we are going to do is something similar against their value for 1050 which happens to be 1050 minus 1262 divided by 118 This is minus 1.7966 minus 1.7966 I will again find the P value for this one point 7 19 6. This is 0.362 0.362 So this is this is 0.36 to 0.362 So I would say that this is the fourth percentile, right? This happens to be the fourth was indicted. These are our answers.