The data from exercise 3 follow. (xi)/(yi)

step 1 Here's the solution to problem number 25. We're supposed to find the standard error of the estim

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It involves estimating parameters that describe the relationship, allowing predictions and inference about the influence of predictors on the response variable.

Standard Error of the Estimate

The standard error of the estimate measures the typical distance between the observed values and the values predicted by the regression model. It is a key metric to assess the accuracy of predictions, reflecting the variability in the dependent variable that is not explained by the model.

T Test for Regression Coefficients

The t test in regression analysis is used to determine whether an individual regression coefficient is significantly different from zero. This test assesses if there is a statistically significant linear relationship between the predictor and the outcome variable, by comparing the estimated coefficient to its standard error.

F Test for Overall Regression Significance

The F test in regression analysis evaluates the overall significance of the regression model. It tests whether at least one predictor variable has a non-zero coefficient, thereby determining if the model provides a better fit to the data compared to a model with no predictors. This test is essential for understanding the collective explanatory power of the independent variables in the model.

Transcript

00:01 Here's the solution to problem number 25.

00:02 We're supposed to find the standard error of the estimate, which is the square root of mse, which is the square root of ssc divided by n -minus 2, and of course, sse is the sum of the residual squared.

00:11 So let's go to our graphing calculator, go to stat and edit, and go ahead and type in your data values in l1 and l2.

00:18 So 26, 9, 13, and 20 is l1, and then 7, 18, 9, 26, and 23.

00:24 Those are your y values.

00:26 Okay, so we need the regression equation.

00:28 So if you go to stat and then calc, and then let's go to that under that eighth option where it says lin reg a plus bx, and then go ahead and calculate.

00:37 And there's your regression line, so it's pretty nice, so it's 7 .6 plus 0 .9x.

00:42 So we'll need that, so let's go ahead and write that down.

00:45 So that's y hat equals 7 .6 plus 0 .9x.

00:53 Okay.

00:54 So now we need to get the y hats.

00:56 So if we go back to stat and edit, and we're going to use this third column here to get the predicted wise now.

01:06 So if we go to use our regression line, we type in 7 .6 plus 0 .9 times l1.

01:16 And those are predicted, and then we can go ahead and find the residual squared.

01:21 If we just take parentheses, l2 minus l3.

01:29 And then we square that squared and we're going to go ahead and add these up so if you go to stat and then calc and then one bar stats and we're going to look at l4 and calculate so the summation of that that's the summation of the residual squared is 127 .3 okay so that's what we need so the s is equal to the square root of the sse, which we just found, to be 127, whatever it was, 0 .3, so 127 .3, and then divided by n minus 2, and n is the sample size, and there were five pairs of numbers there.

02:08 So that's divided by three.

02:09 So you type that in to your calculator, and you should get about 6 .5141.

02:14 So that's the first answer there.

02:16 That's the standard error of the estimate, 6 .5141.

02:23 All right, so part b, so that was part a.

02:28 I'm right, that's part a.

02:29 And then part b, it says to find the t value, or to test the hypothesis using the t test.

02:36 So we need to find the t value, and we're going to use b1 over s sub b1.

02:40 All right.

02:41 So b1, we know is 0 .9.

02:44 That's the actual estimate.

02:46 But the standard error of b1, we don't know what it is.

02:49 So we need to find that.

02:51 So the s sub b1 is equal to s divided by the square root of the summation of xi minus x bar squared.

03:05 All right, i already found s.

03:07 That was the 6 .5141, but i need to get the summation of xi minus x bar squared.

03:14 So i go back to the calculator.

03:16 Let's go back to stat, and then we'll go ahead and find the x bar first.

03:19 So remember the x values were an l1.

03:22 So the x bar is 10.

03:24 So let's write that down, the x bar is 10.

03:28 So we'll need that.

03:29 And then now we can go to stat and then edit.

03:34 And let's use this fifth column now.

03:38 And we'll use parentheses l1 minus the 10.

03:45 So this is the xi minus x bar, and then we square it.

03:50 And we're going to add these together, which means we can just go to stat, calc, of our stats and we'll just change it to 05 now and that summation is 190 so that is this denominator here so we're going to do s which is the 6 .5141 divided by the square to 190 so that's the summation of x i minus x bar squared and that should give you about one point uh actually i'm sorry that was not one point so erase that it was actually 0 .0.

04:28 4726.

04:29 So that's this number here, 0 .4726.

04:33 And we, whenever you plug that in, you should get about, that's your 1 .9044.

04:43 Okay, so that's your t value.

04:45 So we can use that to get the probabilities.

04:48 This is actually a two -tail test with a hypothesis, and so that's 1 .9044.

04:54 So we're going to find this probability, and then we're just going to double it to get this probability over here.

04:58 So we do tcdf, using that 1 .9044...