Question 2. (25 points) A linear regression applied to 23...

Question 2. (25 points) A linear regression applied to 23 data points resulted in the values ?? = 27.5 and ?? = -1.13 for the y-intercept and the slope, respectively. The standard error (s.e.) of the regression line sigma hat (??) = 6.11 and the variance of X = 7.3. Note: Sxx = Var(X) * (n-1) and Syy = ??² * (n-2). a) Compute the t-statistic of the slope parameter ?? to test whether it is significantly different from b? = 0. b) What is the critical t-value to which you compare this t-statistic at the 5% level of significance? c) What is the approximate p-value of this t-statistic? Is it significant at the 5% level? d) What is the coefficient of determination R² for this regression? e) What is the correlation of the X and Y values in this regression? R = ??? ?(Sxx ? Syy) for a fixed value b? of interest are tested with the t-statistic t = (??? - b?) ? ?(??² ? Sxx)

Transcript

00:01 We need to compute the test statistic.

00:03 So the test statistic is given by the formula provided in the question b1 hat minus b1 which is the value that we are testing which in this case is 0 divided by the square root of sigma square hat divided by the sum of squares for x.

00:26 So plug in these information we have the beta 1 or the slope is equals to this.

00:31 Minus 0 is beta 1 either way, then sigma here 6 .11 is square.

00:41 And we have the s, the sum of squares for x, is the same as 7 .3 square times n minus 1.

00:56 So basically what we have here is the n is given to be 23.

01:04 So n is 23.

01:07 So the sum of squares for x is 7 .3 square times 22.

01:14 So if you plug this information here and if you solve this or compute this, you're going to get the slope or the test statistic for the test for the slope.

01:28 Is minus 0 .33437.

01:34 Now for item b, we need to compute what is the critical value.

01:38 So because we are testing if the true beta is equal to zero versus if it is different than zero, because of this difference, we have a two -tail test.

01:52 So when we have a two -tail test, we have that the critical values here are like we have, two, one positive and the other negative, and we just need to find one of them.

02:06 So basically, the idea here is, because we are using, in this case, a regression with two parameters, the intercept and the slope, this means that the distribution that we are assuming for beta is a t student here, and the t student has degrees of freedom for this specific case equals to n -minus 2.

02:29 So in our case, 23 minus 2, which is 21.

02:35 And now we should consider the significance level that we are assuming, which is 5%.

02:42 So with this, we can find, for example, the positive value of the test of the critical value, which is given by the value here, like this critical one, that i'm going to call critical 2, which has an area right to it equals to alpha divided by 2, which is 2 .5%.

03:05 So with these degrees of freedom, and this percentage, as the area, right to this number, we can use this t -table to find that this value is 2 .0796.

03:20 So in the end, the answer for this question will be that the critical value are the positive and negative 207 .9.

03:30 Now for item c, we should compute the p value and say if we can say that this is significant or not.

03:41 So the p value here is always given by a probability.

03:44 Because we have a two -tail test, the p value is twice the probability in a t student distribution with that amount of degrees of freedom that we used in the previous item.

03:59 Or being greater than the value of the test statistic without the sign.

04:05 So 2 .079, oh, sorry, 2 .34.

04:13 This is coming from item a, 34, 37.

04:19 So now again, we can use the t table to find that this here is equal to 0 .029 and 0.

04:29 Which is indeed smaller than 5%, which is our level of significance.

04:35 So we can say that it is indeed significant here when we consider like 5%.

04:44 So this will be the answer for this question, the value of the p value and also that is significant...

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