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1 0 2.23T 1 Hornework: Isino 4 95 2 Sec ccmdcnco 7.2 1 : 1 Confidence conlidence 1 complete} Interval 1 for the colleyose Mean MH 1 Score: 57,5856, 1| 1F Encr YOUr ...

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1 0 2.23T 1 Hornework: Isino 4 95 2 Sec ccmdcnco 7.2 1 : 1 Confidence conlidence 1 complete} Interval 1 for the colleyose Mean MH 1 Score: 57,5856, 1| 1F Encr YOUr anster tne comeids8 111

1 0 2.23T 1 Hornework: Isino 4 95 2 Sec ccmdcnco 7.2 1 : 1 Confidence conlidence 1 complete} Interval 1 for the colleyose Mean MH 1 Score: 57,5856, 1| 1 F Encr YOUr anster tne comeids 8 1 1 1



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Construct the confidence interval for $p_{1}-p_{2}$ for the level of confidence and the data given. (The samples are sufficiently large.) a. $80 \%$ confidence, $$ \begin{array}{l} \mathbf{n}_{1}-300, \hat{p}_{1}-0.255 \\ \approx_{2}-400, \hat{p}_{2}-0.1 Q 3 \end{array} $$ b. $95 \%$ confidence, $$ \mathrm{w}_{1}-3500, \hat{p}_{1}-0.147 $$ $$ \mathrm{w}_{2}-3750, \hat{p}_{2}-0.131 $$

The following is a solution number one, and we're asked to find the 95% confidence interval for the difference of two population means given this data. Now, these sample sizes are less than 30, which means we need to use the tea interval, the two sample t interval if at least one of these is less than 30 and they both are in this case. So I'm gonna use the society for here. And if you go to stat and then tests and it's the show you here is the two sample TNT's. So if you just press zero or scroll up or down to two, samp a T ent click on that and then just make sure stats is highlighted here, you can see your data. So X one bar S for the standard deviation sample size and so on and then the sea levels 10.95 not. You get to this thing called pooled and for this section it's always gonna be yes, because it actually says in the in the very top directions, assume that the standard deviations are the same population. Standard deviations are the same. So whenever that's the case, then you're going to have a pooled, So you just click Yes there and then we calculate and that gives us 16.16-21.8. Force, that's our confidence intervals. All right up here. So 16 .161 to 21 point 839, So that's Are 95% confidence interval from difference of two population means. And then similar here, we find the 99% confidence and we're using this data so again we're gonna use the T interval because these sample sizes are so small and we don't know what the population standard deviation is. So go back to stat tests and then like I said, it's the zero option and we're gonna type in our stuff here. So X one bar is 25. This standard deviation was one, The sample size for the first sample was only six, and then the X- two bar was 17. And then that standard deviation was three and then that sample size was 12 and we're asked to find the 99%ile. So .99. Again we're back at yes and calculate and that gives us 4.2767-11.723. Okay, so let's write that down. Four point 2767. groups All the way up to 11 7-3. So that's The 99% confidence interval for the difference of two population means. Using the TI 84 calculator

The following is a solution to number three and we're asked to find the 99.9% confidence interval for the difference of two, population means given the following data here, and we're gonna use the two sample t interval, because one of these sample size is not at least 30. So in order to use the Z interval, we either need to know the population standard deviations, which is not usually very typical or at least one of the sample size is less than 30. And that's what happens here. So, I'm gonna use the T. I 84 here and if you just go to stat and then tests and then we're gonna scroll all the way down to the zero option where it says to samp T and that stands for two sample T interval, click enter and make sure stats is highlighted, and then you're just gonna pop in your data there. So there's the main standard deviation sample size for the first sample mean standard deviation sample size for the second sample And the confidence level is .999, that stands for 99.9%. And then under pulled we're gonna keep that as a yes, because it says that we're assuming that the standard deviations are the same. So whenever that happens then we're just gonna assume that it's a pooled um appalled uh huh uh what do you call it? Topic or experiment? And when we calculate, we have .133 and .467. So that's our confidence in the rule. So we're 99.9% sure that the difference of the two means between 133 And .467. Okay, so that's our confidence interval there. And then the second Part of this is a 99% confidence interval. Same thing. We're going to the t interval because that sample sizes less than 30 at least one of them is so we go to stat and then tests and it's the zero option And so we need to actually change the stuff. So 77.3 is the mean 1.2 is the first standard deviation and that sample size was 1875 was the second, I mean 1.6 was the standard deviation and then 32 was that sample size and we want to be .9999 Confident there. And again we're gonna keep it as a yes pulled and we calculate so that's 1.1377-3.46-3. Okay, so one point 1377 234 6 to 3. So that's our 99% confidence interval for the difference of two, population means

The following is a solution to number one. And we're asked to find a 90% confidence interval for the difference of the two population means given this uh sample data. So we're gonna use to sample Z interval because the sample size is so large. Whenever the sample size is greater than 30 that's kind of the magic number. The Z. And the T interval are essentially the same thing. So we're just gonna use the tea interval and the way we do that on C. I. T. Four, if you go to stat and then tests, it's this ninth option down here, the two samp Z. And that stands for two samples. The intervals click nine. And then under the summary stats, make sure this is highlighted. Now we're given the sample uh standard deviation but this one asked for the population standard deviation to just put the sample standard deviations here for signal one sigma, two, your ex bar, you're in For the first sample and then your X. Bar, and then you're in for the second sample the sea level, since this is a 90% confidence. And what we'll do .9 and then we calculate and that's about 4.2 and 5.8. Let's go and write that down. So it's going to be ill Not round just so we have a very accurate answer 4.19 6 1 all the way up to 5.8039. So that's the 90% confidence interval Using that data and then the second part, it's the same thing. But this time it's the 99% confidence interval and then we have different data down here, we're still going to use the Z ah the two samples the interval because that sample size is large enough where it basically converges to the T interval. So we'll go to stat and then tests and it's the ninth option. Now we'll have to actually change these. So the S one which is signal one was nine. The S. Two which in this case will be sigma two is four And then the X one bar is negative -112. And then that sample size was 30 and then the X two bar was negative 98. And then that sample size was 40 And then that sea level. Remember we want to do a 99% confidence interval. So 99 and then we calculate And that gives us this here. So negative 18.54 and negative 9.465. So let's go and write that down. So parentheses negative 18.54. All the way up to -9465. So that is our two samples, the interval for these two data sets

The following is a solution for number two and we're asked to find a 90% confidence interval for the difference of two population means given this data now, since the two sample sizes are less than 30 or at least one of them is less than 30 we have to use the two sample t interval instead of the z interval. So I'm gonna use a graphing calculator here. If we go to stat and then test and we are all the way down to the zero option where it says to samp T and that stands for two sample T interval and then we click enter and then just make sure stats is highlighted and we punch in our data here, so X one, bar, s one in one in the next two bar S two and into so that's all the data that was there and the confidence levels nine for 90%. Then you get to this pulled and just make sure it's yes, because it says that that assume that the two standard deviations are the same. So whenever that's the case, then you can go ahead and pull them and then we calculate, and we get this 11.371 to 16.6 to nine, It's just gonna write that down. So 11 .371 To 16, 6- nine. Yes, that's the 90% confidence interval. And then similarly we're gonna do a 99% confidence interval, still using the t interval since they're both Or at least one of them is less than 30 with the sample size. And so if we go to stat and then tests and then just go to the zero option And we need to change this data here. So 68 was Rx one, bar eight was Rs one and then 14 was our first sample size and then uh 43 was our second sample mean three was our standard deviation and 20 was that sample size? And we want to be 99% confidence. And again, we're gonna be pooled. So we say calculate and we get 19.657 to 30.343. Okay, so 19.6 57 2, 4 3. So we're 99% confident that the difference of the two means Given the sample data is between these two numbers.


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