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Nineleen different videp gamos showing alcoho denali and Inal its value?obsenred The duration hmesalcohoseconosreconed When using chis samdoWlestolne claimihaDonula...

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Nineleen different videp gamos showing alcoho denali and Inal its value?obsenred The duration hmesalcohoseconosreconed When using chis samdoWlestolne claimihaDonulation Meangrealer Ihanwhat does dfWhal doe&cenoletThe number degrees of freedom The sample standard deviationThe test etallstcnc sumplc SizlIlne Yaluedfis(Typo = integur 0r J dccmal VO nolrouno;

Nineleen different videp gamos showing alcoho denali and Inal its value? obsenred The duration hmes alcoho seconos reconed When using chis samdo Wlestolne claimiha Donulation Mean grealer Ihan what does df Whal doe& cenolet The number degrees of freedom The sample standard deviation The test etallstc nc sumplc Sizl Ilne Yalue dfis (Typo = integur 0r J dccmal VO nolrouno;



Answers

Was used to complete a $t$ -test of the difference between two means using the following two independent samples. $$\begin{array}{lllllllll}\hline \text { Sample 1 } & 33.7 & 21.6 & 32.1 & 38.2 & 33.2 & 35.9 & 34.1 & 39.8 \\& 23.5 & 21.2 & 23.3 & 18.9 & 30.3 & & & \\\hline \text { Sample 2 } & 28.0 & 59.9 & 22.3 & 43.3 & 43.6 & 24.1 & 6.9 & 14.1 \\& 30.2 & 3.1 & 13.9 & 19.7 & 16.6 & 13.8 & 62.1 & 28.1 \\\hline\end{array}$$
a. Assuming normality, verify the results (two sample means and standard deviations, and the calculated $t \star)$ by calculating the values yourself. b. Use Table 7 in Appendix $B$ to verify the $p$ -value based on the calculated df. c. Find the $p$ -value using the smaller number of degrees of freedom. Compare the two $p$ -values.

Okay. So we know the general formulas are as follows. D mean squares. Yeah. Mhm. Mhm. For both error and treatment is simply the sum of squares comfortable. Mhm. Over the degrees of freedom. Mhm. Okay. Okay. Yeah. Okay. Okay. So this for both the air and treatment we also know the F test formula after statistic from a yeah there's simply T. M. S. T. Over the MSC for the treatment. So now using this these from this you can go ahead and fill at the table. So the mean squares For the treatment is 387 over to damn squares over the use of freedom. Which gives us 193.5. And the F test statistic will be 8,042 over 27. Which will give us to 97 .9. Enough DFC statistic you simply do 1935 Over to 97.9 To give us 0.650. And the total simply something of these two values respectively. To give us 8429 and 29.

Brooklyn 15. Good to me, it's note is that new one is equal to you too. And each one is equal so that new one is not equal to Muto. So the degree of freedom is equal to and one plus in tow minus 2 26 51 minus two people 55 the critical where you are in the room with degree of freedom 55. I'm all for equal toe. Appoint one to tail off table 5. 30. They have two possible negative 2.668 eso the pool standard deviation, which is square. Hold on in one in one minus one times s one squared waas in the tu minus. One times is two squared over in 1% to minus two, which is 26 51 minus two, which is 11.5444 So they remind that s the statistics T is equal to x one minus x two, which is 1 to 7 on this 117 over 11.5444 square root all one over n one plus one over end to which is equal to 3.2573 So at 3.5 2003 is bigger than 2.66 eight. So we reject the non hypothesis, so there is sufficient evidence to support.

So we're going to use the applet that accompanies your textbook, specifically the chi square probabilities and Qantas output. And for part a we want to determine the probability that why is greater than the expected value of why We know that, why is going to have a chi square distribution? And we're going to do this three different times when the degrees of freedom would be 10, the degrees of freedom being 40 and the degrees of freedom being 80. So when you're talking in terms of expected value, you expect average. So therefore we can rewrite that to say the probability that why is greater than average or the mean. And according to theorem 4.9, when you are dealing with a chi square distribution, the average or the mean of a chi square distribution is equivalent to its degrees of freedom. So therefore we could rewrite this and say the probability that why is greater than the degrees of freedom. So we're going to do this three different times. The first time we're going to do this, we're talking in terms of a degrees of freedom equivalent to 10. So we're going to open up that applet and you're going to see a chi square distribution and in the top of it in the curve, you're going to see a place where you can fill in the degrees of freedom and at the bottom you're going to see a place where you can feel in an X. Value and or a probability value. So we're going to fill in the degrees of freedom to be 10. So it's going to now move the Bar around and we're gonna fill in the x value being 10. So what's gonna happen is you're going to see the area here being shaded in And the area is going to be the answer to our question. So the probability that why is greater than our degrees of freedom of 10 is going to be what fills in in the probability spot, which is going to be a .4 For zero. So the answer here is .440 and it's referencing the area in this right tail. So for the second part of this we are going to Look at the applet and do a degrees of freedom of 40. So again, we're going to change our degrees of freedom. We're going to fill in that box, we're going to fill in the X box and we're going to let it calculate our probability box and we're going to get another bell shaped or sorry, another skewed right curves a little bit more bell shaped this time and our degrees of freedom, It's going to be 40 and we're trying to calculate the probability That why is greater than 40. So we're going to put a 40 in the x location and you're going to see slightly to the right of the peak Is going to be where 40 is. And we're again referencing the area to that right? And this time you're going to see a .470. And then the third time we're going to do this And this time we're going to change our degrees of freedom to be 80. So we're gonna have our curve again And in the degrees of freedom box we're gonna type it right in an 80 Where the Xbox Is. We're gonna write in an 80 and we're gonna let it calculate the probability. And this time our curve is going to be very symmetric and slightly to the right of the peak Is where we're going to have our 80. And you're gonna see the area shaded into that right And the box for probability is going to be filled in with the value of .479. So therefore the probability that why is greater than 80 is .479. So part B is asking you to look at these different graphs, Look at the different values and what do you notice? What do you notice about that probability of why being greater than it's expected value or in this instance greater than its degrees of freedom? So what you should have noticed for Part B is that as the degrees of freedom increases? Mhm. The probability is increasing. So take a peek. We've had a probability of .40 when the degrees or 440 in the degrees of freedom or 10. We got 470 when the degrees of freedom We're 40 and we got 479 when the degrees of freedom was 80. And then part C. How does what you observed in part B relate to the shapes of the chi square densities that you obtained in the previous problem? So in that previous problem, you noticed or you should have discussed that the density function is becoming more symmetric. So as the degrees of freedom is increasing. Yeah, the density function graph okay. Is becoming more symmetric. Yeah. It's actually approaching the normal distribution. And if you think about the normal distribution, the normal distribution, the average is smack dab in the center and we've got a Probability of being greater than average of 50. So the probability is approaching 0.5, We're .500. And you could see that again as we look at the three answers from part a. We went from 440 2.470, So you can see as the degrees of freedom are going up, the probability is approaching that .500


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