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(3 points)Two random samples are ectedirom IWO independent populaticnssummary of the sample sizes, sample means and sample standard deviatons given below;55.6, 81 =...

Question

(3 points)Two random samples are ectedirom IWO independent populaticnssummary of the sample sizes, sample means and sample standard deviatons given below;55.6, 81 = 5.2 75.9, 10.1Find 95% confidence interval for the difference /1 Hte of the means; assuming equal population variances:

(3 points) Two random samples are ectedirom IWO independent populaticns summary of the sample sizes, sample means and sample standard deviatons given below; 55.6, 81 = 5.2 75.9, 10.1 Find 95% confidence interval for the difference /1 Hte of the means; assuming equal population variances:



Answers

Find the $90 \%$ confidence interval for the difference between two means based on this information about two samples. Assume independent samples from normal populations.$$\begin{array}{lccc}\text { Sample } & \text { Number } & \text { Mean } & \text { Std. Dev. } \\\hline 1 & 20 & 35 & 22 \\2 & 15 & 30 & 16 \\\hline\end{array}$$

Okay. And all hypothesis, population standard deviations are the same alternative, is that they are not the same or different, does not equal. We're conducting this experiment at a 1% significance law And f is going to be 74.8 squared. Yeah, Divided by 30.4 Squared. This is approximately equal to 6054. Our degrees of freedom are going to be six and eight. Yeah. Mhm. Okay. Since his point all one Half of that is going to be .05, So f. 05. Since it's the two tail test Is going to be 7.9. What just happened there? 7.95. Now, f of 1 -15 or point 995 is equal to one over 10.57. You can find these values in Table gate. Keep in mind that this value right here was found by reversing these values right here in table bait, 1/10 57 is equal to 0.09. Yeah. Alright. Now that we've got all these numbers here, we can compare these two numbers there. So for two tailed tests, we got two tails as the name says, Okay, I don't really know how to draw this, but let's just go with this shark fin looking thing. So we have 7.95. So that's going to be right here. Mhm. Yeah. So that's gonna be the 7.95. We'll just call that eight And the other one is your .09. It's gonna be over here. We'll call that zero. And then we got six which is smack dab in the middle. Therefore At the 10 Sorry, significance level. We failed to reject the null hypothesis. Whoops. Can't do that. There we go. Now. For the beautiful part that I like the confidence and rules. Here they are. Mhm. I believe we figured out our values from last time 7.95 and 0.09. So all we have to do is plug 7.95 in right here, 10 0.09 Right There. And the corresponding standard deviations For the lower bound will get 0.873 For the upper bound will get 8.202. Therefore our 99% confidence interval is 0.873 two, 8.202.

We want to construct the confidence interval about the population mean, you given a sample with X bar equals 25 for sample mean and a number of sample objects 36 sample standard deviation as equals three. Where the confidence level is 95% of alpha equals 950.95 So to start, let's identify the T score needed for this particular confidence interval. The T score is found in a tea table which maps for the degree of freedom in this case 35 and the confidence in this case .95 are exactly score needed. Doing so we obtain TC equals 2.03. Next, let's calculate our margin of error E, which is T C times sfr root end plugging in these three variables for this problem we obtain equals 1.2 Next let's find the confidence interval which is simply xy minus E. Is less than new. Is that the next part plus E? Which gives us confidence interval 23.98 is less than you, is. Less than 26.02.

Okay, so in this problem here, it has given us are mean, Which is our export is 25, Our sample size N is 36, Our standard deviation is three and we have a confidence at level, sorry, a confidence level of 95%. And we want to figure out what is the confidence interval for this information here? So a confidence level of 95 gives us a Z score value of 1.96. Yeah. And then we plug it into our formula, so we have x, Y are plus or minus my Z value times my standard deviation divided by the spirit of my sample size. So I'm gonna plug in my values. So I have 25 plus or minus My z score, which is 1.96 times My standard deviation was just three divided by the square root of 36. So when I calculate that because of this plus or minus, I will have actually two values. So my negative value, which is my lower bound, is going to give me 28.43 and then my positive, which is my upper bound is going to give me 31.57. And so my contact, my 95 confidence interval is actually somewhere between 28.43 and 31.57. Yeah.

Okay, we're doing a test at the 5% significance level. Our and all hypothesis is that the two are going to be the same Are alternative hypothesis is that We have a 14.5 for s. one and a 30.4 for us to like this. Left hail test easy. Okay, so the next thing we do is calculate S one squared over S two squared. So that is going to be the following here. 14.5. Yeah. And 34. Yeah. Okay. No, this is going to be approximately equal to 0.228. Mhm. Our degrees of freedom or his fouls. We're not doing five and 15. It's going to be 10 and eight Since we have N -1 for both those much. And then we can go back here Yeah. To table eight And the f of 0.025 Which is half of five Is going to be 4.3. And this time around this is D F N and D. F. D. When we go of search for this in Table eight. Uh huh. Yeah. Now for 1 -25 which is 0.975 It is going to be won over 3.85 which is approximately 0.26. But you must be asking yourself wait, Why is it 3.85? Well, we have to switch these. That's just the way a left hail test rolls. Yeah. Okay. So now we can proceed with the rest of our problem. We found out this part. We found out that part. Yeah. Um That sets us up for the next part to conclude part A. We see that it is .228. Uh this is going to be less than 4.3. So just for illustration, I honestly I don't know what the f curve looks like here. So what we have is 4 3. That's our cut off value shaded region. To the left tail. Left tail test .228 gets us well within that rejection region. So we will reject Yeah, you know, hypothesis At the 10% significance level. Not for the next part. It's a little bit more fun. We get to construct a confidence interval. Um This confidence interval level is 95%. So we're just going to be doing this guy erase that every step. So what we have is 0.05 And 0.95. And we got a multi divide these by two were multiplied by one half. So FF .025 is going to be 4.3. Mhm. And f of 095 49 You know 975. So that yeah right there we go. Like this slight correction mm We get the f of nine or 0.975. This is going to be equal to one over 3.85, Which we discovered earlier to be easier to six. But and now we can calculate the other values. Okay for our first Valium it's going to be one over the square root of this guy over here. So one over 4.3 times 14.5 divided by 30.4 Is going to be 0.23 scared And the other one is going to be 0.935. So are 95% confidence acceptable Is going to be from .23 2.935.


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