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Samples of packets of cereal filled by two different machines were examined, and the following results about the mass (x grams) of the contents were obtainedSample ...

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Samples of packets of cereal filled by two different machines were examined, and the following results about the mass (x grams) of the contents were obtainedSample SizeMeanVarianceMachine 340.9 16.82 Machine B 341.4 8.63 Test whelher the mean masses of contents in packets lilled by the two machines are significantly different at the 5% level. Assume that the masses of contents in packets filled by each machine are normally distributed, and that the two normal distributions have equal variances_H

Samples of packets of cereal filled by two different machines were examined, and the following results about the mass (x grams) of the contents were obtained Sample Size Mean Variance Machine 340.9 16.82 Machine B 341.4 8.63 Test whelher the mean masses of contents in packets lilled by the two machines are significantly different at the 5% level. Assume that the masses of contents in packets filled by each machine are normally distributed, and that the two normal distributions have equal variances_ Hint: use the formula Test statistic when sampling from normally distributed populations: population variances known 7) Professor Abena Mensah Microbiology instructor, decides examine (wo classes consisting of 40 and 50 students, respectively_ In the first class Ihe mean grade was 74 with standard devialion of 8, while in the second class the mean grade was 78 with standard devialion of 7 . Is Ihere significant difference between the performance of the two classes at the 0.01 level? Hint: use the formula Test statistic when sampling from normally distributed populations: population variances known



Answers

Cereal The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces. You open a new box of cereal and pour one large and one small bowl.
a) How much more cereal do you expect to be in the large bowl?
b) What’s the standard deviation of this difference?
c) If the difference follows a Normal model, what’s the probability the small bowl contains more cereal than the large one?
d) What are the mean and standard deviation of the total amount of cereal in the two bowls?
e) If the total follows a Normal model, what’s the probability you poured out more than 4.5 ounces of cereal in the two bowls together?
f) The amount of cereal the manufacturer puts in the boxes is a random variable with a mean of 16.3 ounces and a standard deviation of 0.2 ounces. Find the expected amount of cereal left in the box and the standard deviation.

So we're looking at people wearing seatbelts and not wearing seatbelts Children and finding out how many days they stay in the I. C. U. And so we're going to assume that those students with a seat belt have equal stay in the hospital as those who do not wear a seatbelt. And alternately that the seat belted Children have a mean stay less than those who were not seat belted. And so we're assuming that the difference between these two seat belt minus not seatbelt is zero and we're actually getting something that's negative and this will be our p. Value. So let's look at the data and let's get our test statistic and we have sample sizes to sample sizes. One sample size was 1 23 and the other was to 90. So that was the for the seat belt and this is for the non seatbelt. And so we're going to use degrees of freedom of 122 to be conservative. And they're relatively large sample sizes anyway. And so let's find our test statistic and we have our 0.83 minus are 1.39 And then we're going to divide that by the standard deviation of the seat belted group squared, divided by the sample size, and then the standard deviation of the non seat belted Children divided by the sample size of it. And when we do that calculation, we get the test statistic comes out to be negative 2.330 And so now we want to find, and that's so that's what this value is. We want to find the likelihood, if the difference is actually zero or the means are equal, how likely is this number or more extreme to come up? And so I'm going to use my uh my T c D E f to find this mighty CDF. And I'm going to use negative uh minority has negative one times 10 to the 99th hour in it. And then our upper value is going to be that negative 2.330 My degrees of freedom is again the 1 22. And I will paste and let that do the calculation. And I have a value of a p value of about 1%. And that is less than my 5% significance level. So I definitely have sufficient evidence to reject the null meaning that we believe that the Children that had seat belts on have a smaller uh stay in the uh in ICU than uh those that are non seatbelt. And again you write my sons for that. So now we want to find the confidence interval and the appropriate confidence intervals. Since this was a one tailed test and we had 5% in this tale, the confidence interval has 5% down here, 5% up there. We would need to look at a 90% confidence interval. And so my table does not have a T. V. Star value for 122 degrees of freedom. So I'm going to use my inverse T. Value my inverse teeth. And on the inverse T. I want the area below it to the point oh five. So I'm going to put in the 0.5. It will be finding basically this low limit which will be symmetrical. What the upper one and then our degrees of freedom are that 1 22? And we get that value to be Yeah. Uh And this was 1 22 comes out to be negative 1.657 And so now let's get that confidence center, let's look at the difference first. So we need this difference the 0.83 minus 1.39 And that difference comes out to be negative 0.56 plus or minus. And we have that T star value 1.657 because we have one down here and went up here and then we have this same uh same standard deviation we had here. So let's see if I get 1.77 squared and that is a 3.6 squared mm And our sample sizes are 1 23 and 2 90. Okay, And let's get that margin of air, Yeah. Mhm And 1.657 times square root of 1.77 squared divided by 1 23 plus 3.6 square divided by 2 90. And that margin of air comes out to be 0.398 and two. And it keeps going on but I'm just going to store that is X. And so let's get those two values. We have the negative 20.56 minus the X. Value. And that comes out to be negative 0.958 And then we can change that into an addition sign. And we find out that that is still negative and notice that that does not include zero. So we would again from this interval see that the difference does not appear to be zero as he appears to be negative. So it does appear as though um that the seat belts make a difference seatbelts beautiful. They're good that they appear to have a lesser chance of a lesser stay in the hospital in the ICU for Children.

Let us look at this question. Milk consumption. Okay, You are performing a study about weekly per capital milk consumption previously found weekly per capital milk consumption to be normally distributed with this mean and the standard deviation. And we have 30 people and we have recorded their milk consumption. Draw a frequency instagram to display the data. Okay, Do the consumptions appear to be normally distributed? My apologies. Now, does this appear to be normally distributed? Yes, we can say that this is pretty normally distributed. This looks unit model on equally spread almost equally spread on both the sides. So yes, and the mean understanding deviation off the sample. The mean we can see here is 45 point 86 and the standard deviation is 9.46 And compare the mean and the sample standard deviation with those of the previous study. Well, the mean previously was 48.7. And this time the mean is 45.86 So the mean has decreased, and the standard division this time is 9.46 and last time it was 8.6. So this time the, uh we can say that the normal distribution is more spread out and these would be my answers

Okay for this question, The population population equals 36. Sorry. 34 comma. That's a six. Coma, 41 coma 51 on, then. Any costed too? Because too, um, two samples of selected without replacement. We replacements. I'm sorry. So any posted to And then, um, the sample size for the first question, Because the first question save, um, if after identifying 16 possible samples, damage the sample size a 16. Okay, the sample size is 16. And then, um, after I'm going to find that find me off each sample. So we're supposed to find the meaning of example. We already know the population. We know, um, and we know December size, so finally, we don't. Each sample is very easy. This is going to be, um, sample mean because the some off all values divided by the number of the values and then the sample mean is gonna be the sum of all values divided by the number of values, which is gonna be one divided by 16. Because they're 16 possible samples. And this is going to give you zero points. 06 to 5. And then that's the answer. The second question wants us to own Compare the mean off the Oh, wait. We're still in the first question. I'm sorry. We're always supposed to construct the table representing the sample distribution of the sample. Mean and compared. Combined values off the sample mean that are the same. So to start the table, um, we're gonna have to ride the samples over here. And then we have to, um, to to know about the picture. Random. So we're gonna have to sit for everyone. So the samples are going to be 34. Commodity before. Let's focus matters for that's for commodity. Six. 34. There's a 4421 34 51 then moved to 36. That's a 6 34. That's his six. That's a six. We have 56. Just the 6. 34 to 1. And then 36 40 51. Rather. And then we moved to 4 to 1. 41 51 321 Oh, I'm so sorry. 41. 34. Okay, what's the 1 34. 41. 36 41. 41 on 41 51. They're mostly fifties. 51 Wanted to four 51 36 okay. 51 4 to 1 and 51. 51 and then we we're gonna have to find the mean because the question wants us to construct table off the sample destruction of the mean off the sample. Mean So the other parts is going to be, um it's gonna be me and the mean is some values divided by the number of values. And, you know, we have to values. We have two values for each one. So that's corpus. That four divided by two is what's going to give us the mean. And that's going to be 34. Yeah, And that's gonna be tested for for the second one. The mean is gonna be testified. It's a seven points five, 42 points, five. And we keep going down. We keep doing this. We get, though. That's a five. Um, that's a six 38 15 43.5 did 7.5. Yeah, 38.5 4 to 1. 46 42 5. It's right. Um, this is not really clear, so I'm gonna clean. Then we have 43. His life 46 and then 51 and then there are next side is gonna be the probability. And since we know that they're 16 samples, each other probabilities is gonna be on over 16 or otherwise 0.6 to 5. So that's going to be the policy for every single one of them. 2.660 point 06 to 5 Take Andi. That's it. Now combining combining values of example Me the same. Like the question told us to, um, the table will look like this. We have the sample mean over here and the probability in the other side. And then we know the sample means the sample means they're all off. These figures often subs burn. And since the repeated, we're going to list them one once. So we have 54. We have 35 we have 56. There's 7.5. We have this 8.5. Okay, we are 4 to 1. We have 42.5. We have 43 plans. Five. We are 46. Oh, yeah, it's a warm. And the probability the probability off well 34 is going to be 0.6 to 0.625 because it was. It was ill cod once on 35 or clouds wise, um, so is going to be so far. Of course. Why? So it's going to be 0.625 times two, and I was going to be 0.125 That's six Accord once, so it's going to be zero points. There was 6 to 5 at 7.5 or coats wise, so I was going to be zero points, 1 to 5. That's 8.5 o'clock twice to 0.15 41 or called once three points. There was 6 to 5 and then 42.5 or code twice. What's 3.5 or cold twice? 46 or quote twice. Zero planes once or five. And if it's one or cold ones and that's the end of the table. Um, the next question stays okay, compared the means of the population to the mean of the sampling distribution of the sample. Mean so we know that, um, the meaning of the sampling distribution off the sample mean is going to be you mean is going to be the total number off. Um, little talking about off mean off. Sample me. I'm sorry. Sports rooms was right. Sample mean the total number of sample mean divided by the own Well, it's supposed to be like the sample mean the probability multiplied by the sample mean each in each one of them. So 34 multiplied by zero points 06 to 5. 35 replied by all that. But the final answer is going to be footsie. So basically, we're gonna dio basically, we're going to do 34 multiplied by 0.6 to 5 on multiply all of that to you And they asked, The hotel on is going to give you 40.5 and thus the mean or you can do it. You can do it by saying on zero points you can do it by saying 0.6 to 5 on times 34. Um or you can do it by saying 3.6 to 5, then opening in brackets, added all the figures that has your 50.6 to 5 to make the calculation easier. But whatever way you choose to do it, you're still gonna get 40.5 as the meat off the sampling distribution off sample mean? And then you want us to compare thio mean of the population, the meaning of the population, or rather, population Me. Okay, publish on me population mean is the not the total number off the the some off the figures, which is the sum of the population, which started for 36 41 51 divided by the sum divided by the number. That's four because their floor populations. So it's gonna be that. Is this for 34 waas 36 close 41 Nos. 52 divided by four. And then this is going to give you 40.5, 14.5 and you can see they are the same. So they're equal. And then the next question three. Next question says, um, does that mean does this sample means that gets the value of the population mean under the sample means make good estimators of the publishing mean that so we yes, for both of them, the sense the sample mean targets the population mean because it is an unbiased estimator off population mean and it's because they're equal because your equal yes, yes,

So we would be assuming that the body mass index mean for females is equal to the body mass index for males and alternately that they're not equal. Do we just want to find out whether they are the same or not? And so we're going to be assuming that the difference of put female minus mail is zero and we're actually getting a difference that's something positive. And so we're getting something up here. But Again, we're doing a two tailed tests. So this is the actual difference. So let's find the test statistic and we can see that will do the conservative measure and not use the formula for degrees of freedom. But we would say the lesser sample sizes 70, so less one is 69° of freedom. And we need to take that 29.1 -28.38. And then we're going to take that divided by the square root up and that 7.39 square divided by the sample size of 70. And the 5.37 squared divided by the sample size of 80. And when we do that we get a test statistic of .674. And so this corresponds to a T. Value of point 674. And this one is negative .674. And so let's find the actual P value instead of finding the critical Z. Value. That or T. Value that would be for the rejection. So let's find the likelihood of getting a test statistic like this or more extreme. And that's for this tail and then we'll double it to add in this tale. And I used my T. C. D. F. On the calculator and when I did I got that p value to be about 0.50 about three. And so our significance level was five and this is not less than 5%. So we we fail to reject the null even though the females is higher. We failed to reject the null. This is not considered to be significantly different. This difference between them is not considered to be significantly different. So the main seem to be basically the same for the BM body mass index. Now, we wanted to find a confidence interval that would be appropriate for this. And since our significance level was 5%,, We would be finding a 95 competent centerville. Now, if you look up in your table, you most likely do not have one for 69° of freedom. In my table. It shows 60 and 80, but I can use my inverse T. In order to find that. And that's what I'm going to do. I'm going to go to second and distribution and I'm going to get that inverse t. And I'm actually going to plug in an area of .025. So I'm gonna, with that confidence interval, I want the area here to be .025. And that to be .025. And then that gives me 95 in the middle. And then I'll put in for 69° of freedom and let me type that in. And I'm getting this negative one when I put it in this way, Yeah. And 69° of freedom. And so when I do that, I get that that value comes out to be negative 1.99 basically five. So when I find that confidence that are only going to put the difference, I got plus or minus that 1.995, that's my T. star value. And then I'll have that square root here. Now let's see what we have as our numbers. 29.1 and then the other difference or the other value for the males. Was this? And so let's write what that is. 29.1 -18.38. That difference is only .72. I don't know what unit they put on B. M. I don't know what the I guess it's just a number. Mhm. I have a number of comparison. I don't know that's percent or anything like that. But could be And then we have that 7.39 squared here divided by the sample size and that 5.37 sq divided by the sample size. And so let's find that margin of air 1.995 times the square root of. And we'll play pin that 7.39 squared to fight it By the 70 Plus the 5.37 squared Divided by the 80. And that margin of air comes out to be 2.13. And so now I'm going to store that value is X. And I'll take that .7 to minus my ex value to get that lower limit. And that's negative 1.41. And then when we add well we'll just look at that, that's going to be 2.85. And so we think the difference with 95 confidence is somewhere in here and notice that that includes zero and which means we have no evidence to discount the idea that they're equal because zero as the difference lies in this interval. So in part C do we think there's a difference? There doesn't appear to be a difference. No, not a difference in the main B. I for males and females, there doesn't appear to be a difference without those two sample sizes


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