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Consider simple linear regression model Y; = D + BX; + € where & Are in- dependent identically distributed (iid) N(O,&") . Suppose the parameter ...

Question

Consider simple linear regression model Y; = D + BX; + € where & Are in- dependent identically distributed (iid) N(O,&") . Suppose the parameter values are Bo = 200, 8, 5.0 , and & = 4Explain the meaning of the parameters' Bo 200, Bi 5.0, and 0 = 4 Assume that the scope of the model includes X = 0. b) What is the distribution of Y? What is the mean and variance for this distribution? Display the distributions of Y for X 10, X "20, and X 40. Hint: draw random sample o

Consider simple linear regression model Y; = D + BX; + € where & Are in- dependent identically distributed (iid) N(O,&") . Suppose the parameter values are Bo = 200, 8, 5.0 , and & = 4 Explain the meaning of the parameters' Bo 200, Bi 5.0, and 0 = 4 Assume that the scope of the model includes X = 0. b) What is the distribution of Y? What is the mean and variance for this distribution? Display the distributions of Y for X 10, X "20, and X 40. Hint: draw random sample of 100 using rnorm() and display the histogram using hist () commands in R, d)* What is the sample mean and the sample variance for the random sample of 100 in part (c)? Compare estimates with the population mean and the population variance:



Answers

A population has a mean of 400 and a standard deviation of $100 .$ A sample of size 100,000 will be taken, and the sample mean $\bar{x}$ will be used to estimate the population mean. a. What is the expected value of $\bar{x} ?$ b. What is the standard deviation of $\bar{x} ?$ c. Show the sampling distribution of $\bar{x}$. d. What does the sampling distribution of $\bar{x}$ show?

All right, That's the question. 1 83 So in this question, our simple size and IHS 50 are a We moved to the question part So part A what is a no hypothesis? So here the now hypothesis iss the population mean the population mean is 100. And if we use me to represent the population means so our no hypothesis is going to be your equivalent to say, um us Equal Teoh 100. All right, Part B, what is the alternative hypothesis? So here we can say the population means is not 100 or we just say mu does not equal to 100 note. Here it is a not be close eye. So when we considering the peat p value winning to consider two tails together, just keep in mind. All right, heart, see what's value for Alfa here in the question it is stating that using a airplane there a one level of significance. So that means our Alfa is going to be there a point there or what? All right, um, de what is a hypothesized, uh, value from you? So has wasted in the part a, um in the no hypothesis, our hypothesized highly valuable for me was going to be 100. Okay, part A what? Ihsaa value for ex parte um, export. Here it ISS represents the Sam Pony, and it is given in a question. On the better is 96. All right, heart f What is a standard deviation? They're here. Siegel eyes representing the standard deviation TV's. Okay. And, uh, and as equal to 12 partying siga off export. Here it is representing the standard arrow center era off. I mean and by the formula Friends, it is equal to speak on Monday, but Abay Square showed up in, so it's going to be 12 over square. Oh, no. 50. It is equal to 1.7 Terrible. All right. And we moved to h. We're interested in the testes. Statistic is that star. So we need to do some calculation here. According to the former learns that that start is equal to x bar minus the hypothesized value over mu over Segolela divided by square enough. And so we just plug in numbers. We have 96 minus 100 divided by 12 over Square Rudo 50 and it is equal. Teoh Negative 2.36 All right, what is the What is a P value here? The P value is going to be the probability off a random bearable that which absolute value is greater than or equal to the absolute value of ours that start Or, in another words, it issue. Go to the probability off a stunner. Normal, um, random bearable. That which ISS smaller than or equal to negative 2.36 So this is our of value for its that star. Plus the probability off is that greater than our equal? Teoh? Positive 2.36 All right, if we draw occur, representing the center normal distribution in here and it is going to be negative. Teoh calling 36 and symmetrical way Here it is. Positive 2.36 So our Peabody, who is going to be the summation off the area for this Tau and the sheriff of this tale as well? The ended up and equal Teoh the P value. So this, um, south is equal. Teoh Narrow point narrow 18 All right, Pant heart J wants a physician. So we were going to compare p value with our level of significance with the Alfa I recall Alfa is U Boat Teoh. There a point there. What? And hear? The P value is equal to their opined there. 018 And obviously our people I always greater than Alfa So here we are fell Teoh reject the null hypothesis.

In this exercise were given data for two populations. We have I mean for 7.9 for the fast population and the standard deviation of 5.4. Also given I mean of 7.1 for the second population and the standard division of 4.6. So we're supposed to I use this theater to find the mean and the standard deviation of all possible differences between the two sample means. So that's the first part of the problem. A And we are also given that the sample sizes will be three and six respectively. That means n one equals three and and to equals six. So first we get the mean of all possible differences between the two sample means. And that's given as follows, mu X r one minus X box too, which is obtained from the formula new one minus new too. So we use the values, we substitute the values into the equation and we obtain 7.9 minus 7.1 and that equals 0.8. Next you find the standard division of all possible differences between the two sample means and that is denoted fine sigma X one bar minus x two bucks, which is obtained by the formula square root of sigma one squared over n one plus sigma two squared of an end to now. When you substitute the values we have the following. You have five 0.4 squared over three. Loss 4.6 squared over three. When you work that out, you obtain the following, you get 9.72 plus 3.53 which equals square root off 13.25 which equals 3.64 So that's going to be the standard division of all possible differences between the two sample means. Next in party. The problem, we are checking what are the variable under consideration must be normally distributed on each of the two populations. For the answer that we have given in part A and the answer to that is no. This is because the formulas for the mean and standard deviation of all possible differences between the two sample means hold regardless of the distributions of the variable on the two populations. So the formulas in part A for uh for the mean and also for the standard deviations would hold even if the variable was not normally distributed. Uh in part, see the problem, we're going to be uh determining yeah. Whether we can conclude that the variable X one bar minus X two bar, which represents all possible differences between the two sample means is normally distributed. No. In this case we see no, because it has not been explicitly explicitly stated that the viable uh under consideration is normally distributed. And also the sample sizes are very small for us to use the central limit story, um to approximate that uh the variable is normally distributed.


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