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Blind laste test is condueted (0 determine which of two colas Brund are randomly asked to drink one of the two types of cola first, Brand E, individuals prefer. Ind...

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Blind laste test is condueted (0 determine which of two colas Brund are randomly asked to drink one of the two types of cola first, Brand E, individuals prefer. Individuals prefer; followed diselose the drink they Results of the taste test indicate by the other cola, and then asked t0 0.06 level of that 65 of |00 individuals 'preler Brand Conduct hypothesis test at the ( 'significance t0 see the preference of the two brands is about the same_ Construct 96% confidence interval for the p

blind laste test is condueted (0 determine which of two colas Brund are randomly asked to drink one of the two types of cola first, Brand E, individuals prefer. Individuals prefer; followed diselose the drink they Results of the taste test indicate by the other cola, and then asked t0 0.06 level of that 65 of |00 individuals 'preler Brand Conduct hypothesis test at the ( 'significance t0 see the preference of the two brands is about the same_ Construct 96% confidence interval for the proportion opindivedcals who prefer Brand B What is the Confidence Interval? 13 . What is the test statistics? (0.5520,0.7480) (0.5620,0.7380) (0.5720,0.7280) (0.5820,0.7180) What is the p-value? 10. What is testing? 0,0627 0.0027 0.1027 0.0327 15. What should we do? 14. Two tailed Right-tailed, or Left-tailed? Reject the Ho Two-tailed Cannot reject the Ho Right-tailed Reject the H; Left-tailed Accept the Ho



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(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value(s) and identify the rejection region$(s),(c)$ find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. If convenient, use technology. A company that makes cola drinks states that the mean caffeine content per 12 -ounce bottle of cola is 40 milligrams. You want to test this claim. During your tests, you find that a random sample of twenty 12 -ounce bottles of cola has a mean caffeine content of 39.2 milligrams. Assume the population is normally distributed and the population standard deviation is 7.5 milligrams. At $\alpha=0.01,$ can you reject the company's claim? (Adapted from American Beverage Association)

So we're going to assume that the volume of a regular coke is equal to the volume of regular Pepsi and alternately that the volumes are different. So we're going to do a two tailed test and which means picture wise, were assuming that this difference between and let's say I take regular coke, regular coke minus the regular Pepsi could do the opposite way to. We're going to assume that that means Dick is zero. And then we will look at the actual difference we're getting is point one ounces. And because we're doing a two tailed tests, will also look at the 20.1 and this is really a difference of negative 0.1 and this is 0.1. But these aren't test statistics. These are the actual numbers and these combined together will be R. P value. So we want to find what that test statistic is. And we will use the conservative estimate and say that the degrees of freedom is 35. And we're going to take that 12.129 minus the 12.29 which again, is that 0.1 ounces. And then we will have this square root of our standard deviation square to the first sample divided by sample size and the standard deviation for the funding. The variance basically of that one, and then divided by the sample size. And when we get that test statistic, we end up getting negative 4.22 So we want to find what's the likelihood of having that come up or more unusual. So this value again corresponds to a test statistic of negative 4.22 for a T. Value. And this one would be positive 4.22 And so we're just going to double this probability to find our P value. And when we do this, I use some software to find us. And this comes out to be pretty close to zero, comes up 2.16 So it's very small. And so this is smaller than the 5% significance level. So we would have strong evidence because it's so much strong, so much smaller. Strong evidence to reject the null and conclude that their volumes are different. So volumes are different. That's what we would conclude now for the confidence interval, the confidence interval we would use for part B. We would use a 95% confidence interval because our 5% significance level has already been split into those two tails. Now on my book that I'm using, I didn't look up online. This book my book only shows 30 and 40. So when I look up 95% I find my critical T. Value for degrees of freedom of 30. I don't have a 35. So my 30 is 2.42 and that is 2.5 in the upper tail, 2.5 in the lower tail. Or looking up on your table to look where the 95% confidence interval is. So now I'll do my calculation of the confidence interval. So I'm gonna take that difference which is 0.1 negative plus or minus this value 2.42 And then we'll use that same standard deviation here of the 0.11 square divided by the sample size plus the point oh nine squared divided by the sample size. And so let me get what this whole margin of areas here and I didn't calculate it ahead of time. So I'm going to type in this value first and squared plus 36 plus the 360.9 square divided by the 36. And then I will multiply that by that. 42.42 And so I get negative 0.1 plus or minus and I get 12 point 254 And so when I do that calculation open let me go back. I can see that I have an air that looked way way too big. And I put an addition sign where there should have been a division side and should have double checked that you would have looked at them and said holy cow, that's huge. And then multiply that answer times 2.42 All right. So this is not this is not right. I'd have negative 0.1 plus or minus. I knew that was way too big 0.4837 So let's add it on. I'm going to actually store that value and it has more decimals to it but we'll store that value uh and I have negative 0.1 minus that value which is going to give me negative 0.1 for eight as the lower limit. And then we have that plus that value point want excuse me negative 0.0.1 plus that value 0.48 And we get negative 0.52 and notice that that confidence interval does not contain zero. So we're 95% confident that the actual difference between their volumes is somewhere in here, which means it looks like it's going to be negative. Okay?

Hi all the different portion the americans prepper or or Pepsi. They're saying that the 300 individual wrapper crap pepsi to take another 100 access to victory. Right? So being bored here, Not just basically the americans prefer or Pepsi And we take it up to 300 individuals. Rapper Cropsey. So we take here now before this starts coming activity equals pino and already able to spin or equal to pino. Protect the different ways of pino from looking forward to in order to take from Did a point or two to the romantic sports. And this solution anything five values rest. You can find out accordingly and calculate the p value probability and check the hypothesis which was used to prep for rejecting policies or to uh not trajectory policies, alphabet India Europa, Europa. You get that lonely. Let's find out the probability and it you want to buy. No problem. Any um technology before the property values there is this 0.4 to 0.2580 point florida, p 0.3169 and so on. So what we can see here this is a property values less than alpha. So we can reject not to reject north Korea. Jack Peter Bergen out to not to reject. And this will reject. So you can find all the values. And the final answer is uh coming up before this one that While there's a peanut between Europe 144 and 0.62 european port for and zero point thanks to the peanut values in these values uh will not reject the null hypothesis. Not to to reject that north. Otherwise you reject at north. Then we have been or test representing the population proportion. I want to be back now the marvel kind of government approval for that. Uh huh. Just a moment. Yeah we have here just even asked have a proportion plus minus Z. Critical value. And then this bomb load is we'll find a form the approval. You just find the sigma here. That is the square root of proportional 19 negative reporting over them. Probably 2.5 3.499. Then the biblical values coming out to be 1.96 for retail value. Right? The sound of abortion. And then you have this violence. So we got 253.432 2.6-8 For the sleep part. If you go to hell fire it's called 0.01 Football. This will also not be rejected. It will also not be rejected. But we don't have any values to reject all party. The whole point 01 abortion. This path why does this week friends. Now this makes sense because the as we just go toward lower value of alpha. This means we need more convincing evidence to reject. The only possible. We need more convincing evidence to reject that everybody that called. That's why we reject we expect a larger range of possible values. Barbara population proportions. We need more values for the population proportion to that. We can ejecta people. Thank you.

So we we would be assuming that the mean of the diet coke is equal to the mean weight of the regular coke. And alternately we think that the diet coke is actually less than the regular coke. And we have our data that's given to us in the example of two sample sizes of 36. So really we're assuming that the difference between the two is zero, so that the mean of diet coke minus the mean of regular coke is zero. And then we're going to get a difference that's actually negative. That's what this is our actual difference. And we want to find what this likelihood is and this will be our p value. So let's find our test statistic, our test statistic which we're going to use the conservative estimate and find that 35 degrees of freedom. So we need to know what this test statistic is for for this specific value. So we're going to take our mean of the two, which is that 20.78479 minus the mean of the other, which is that 0.81682 And then we're going to divide that by the square root of And we're gonna take our standard deviation 439 squared divided by the sample size. And again this standard deviation 0.751 square divided by the sample size. And this gives us a test statistic that is negative 22.9 And this is very, very uh Mhm incorrect. As far as where I place that difference. That difference is like way down here in this distribution. So the likelihood of getting this type of test statistic for that difference, Mhm is approximately zero. So we have strong evidence to reject the null. We have strong evidence to reject the null and claimed that the diet coke does have a lesser weight, lesser weight than regular coke. So we have strong evidence of that. That doesn't matter what significance level you use. But we were supposed to use 5% and then it asks us in part B will give us the appropriate confidence interval. If we're going to use this idea. And if we were doing a confidence interval, we would need to use 5% of the lower tail, 5% of the upper tail. We'd actually need a 90% confidence interval to look at that. Not a 95% because we would want 5% of the air at the left tail, 5% at the other tail. And so on my table, I looked at my table and mind skips from 32 to 40. I don't have 35 degrees of freedom. And so I looked for the confidence interval number. So I would have 5% in the upper tail or you would have down at the bottom, it would say a 90% confidence interval and I have this value comes out to be 1.697 Now your book might be more accurate than mine. So if you can get more accurate you would and so we would take the difference that we got and so we would have the point 78479 minus the 0.81682 plus or minus. And then we would use our T star value for that. Now, if we use software will get a different value and then we still use that same standard deviation of the 0.439 squared over the sample size plus 0.751 squared over the sample size. And I'm going to leave that for you to do the calculation of finding what these two numbers are. But we'll find that it basically does not include zero. So these numbers are going to be different now. Part C. Asked why why would this happen? Well, the diet coke uses artificial sweetener versus the regular cope. He uses sugar and this must be more dense. So it's not that you're getting a lesser volume, you're just getting a lesser mass because of the sugar being dancer. Mhm. Okay.

Problem. 16. It's in a each note that new one is one of the record toe. Each one is that new one is bigger than you, which can be a degree of freedom. Equal toe end one plus any to minus two, which is 25 plus 15 minus +23 38. Yeah, the critical values are in there. Always Degree friend of 38 on offer equal toe open or one off table five. So t is equal to 2.449 Oh, there is actually when they contain already greater than 2.42 money. So the old standard deviation in that square rode off and one minus 1 25 minus one time 60 square waas. And to win this one, which is 15 minus one times, is to square with 80 over and one plus any to Minnesota 25 plus 51 less to approximately 68 point or 56 So t is equal to x one of our minus six to voice with 320 minus 280 over a full cent divisions of 68.456 square. Load off one over anyone Waas one over and to which is equal to 1.8. So if the value of the testes is is in the rejection, then they now have got it. Project at 1.8 is smaller than two point 4 to 9. So feel to reject than our high processes, so there is no sufficient evidence to support this screen.


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