[DATA] The following data represent the height (inches) of boy...

Question

[DATA] The following data represent the height (inches) of boys between the ages of 2 and 10 years.

Table displaying boys' ages from 2 to 10 years with corresponding heights in inches across three columns.

b. Compute the standard error of the estimate, Sₑ.

Accepted Answer

Welcome back, everyone. In this problem, the following data represents the weight in pounds of girls between the ages of 3 and 11 years. Compute the standard error of the estimate SE. A says it's 0.63, B 11.68, C 0.74, and D 2.19. Now what do we know about the standard error? Let's start there. Well, recall that the standard error of the estimate is equal to the square root, OK, of the sum of the Y values minus the predicted y valuesy hat squared divided by N minus 2, OK. Where we can get our predicted values of y y hat from the least squares regression line equation which says that it is equal to B0 plus b1 multiplied by X where B 0 is the intercept and b1 is the slope. Now, before we can find these predicted values of Y, we need to figure out what the slope and intercept is so that we can determine the predicted values, OK? So Here we know that our. Uh, or slope B1 is equal to N multiplied by the sum of Xy values minus the sum of X values multiplied by the sum of Y values divided by N multiplied by the sum of Xquared values minus the sum of X values squared. OK? And no. We can use B1. Let me just pull this over a little bit. We can use B1 to calculate or intercept B not because B0 is gonna be the sum of Y values minus B1 multiplied by the sum of X values divided by N. OK? So clearly here, we have a lot that we need to calculate based on the X and Y values. So let's add these to our table, OK? Let's add some columns for all of what we need and then see if we can use that to help us figure out our standard error of the estimate. So like we said earlier, first we'll need uh X squared. And Xy values for us to calculate um our slope and our intercept. OK. So pretty much let's start there for no. No, we have, we already have our X and Y value so we can go ahead and solve. So for example, when our age is 3, then X2, or X value is 3, so X2 is 9, when it's 4 X well let's do the XY value at the same time, so. When X is 3, X2 is 9, and when X is 3 and Y is 32, then X Y would be 3 multiplied by 32 or 96. Next, uh, when X and Y are 4 and 35, then our values would be 16 and 140. OK. Next, we're gonna have 25 and 190. Uh, for 6 and 41, we will have 36 and 246. OK. Uh, for 7 and 44, we're gonna have 49 and 308. For 8 and 48 we'll have 64 and 384. Next, we're gonna have 81 and 468. Uh, 10 and 560. And then 121 and 660. OK? So now remember we'll need our sums, OK? And if we think about it, then let me put this in red below on our table here. The sum of X values is gonna be 63. Y values is gonna be 406, uh X values 501, and Xy values 3,0052. So at this point we have enough information that we need to help us figure out uh what our slope and why intercept are going to be. Because no Put in right here. No, that means first or sloop B1. Is gonna be equal to the number of values 9 multiplied by the sum of Xy values 3,0052 minus the sum of X values 63 multiplied by the sum of Y values 406 all divided by N9 multiplied by the sum of X2red values 501 minus the sum of X values 632. I know when we calculate that we get B1 to be equal to 3.5. Next, this means then that our interceptb not is going to be equal to the sum of y values 406 minus B1 3.5 multiplied by the sum of X values 63 divided by n which is 9, and that is approximately 20.61. So since we have B1 and B not, this means then that or at least squares regression line Y hat is going to be equal to 20.61 plus 3.5 X. So we've made some progress here. We, we have the formula that we need to calculate our predicted Y values. So now, let's go back to our table because remember our standard error estimate equals the sum of Y minus the square root of the sum of Y minus Y values square divided by N minus 2. So we need all our Y hat values thus to get the sum of our residuals. Y minus Y hat. So let's get back here and let's add some more things to our table. We're gonna add a column for the Y hat values, OK. Then our residuals, Y minus Y hat, OK. And then Y minus Y had squared. And now we're going to again, we're going to go ahead and see if we can figure out these values and some all are Y minus Y had values squared to put them into our formula for the standard error of the estimate. Now, starting with uh here when X is 3 and Y is 32, then if we do the math here, Y hat, for example, is going to be 20.61 plus 3.5 multiplied by 3, which is going to give us 31.11. Thus that will give us a residual value of uh 32 minus 31 or sorry. That, yes, 32 minus 31.11, OK. Which is gonna give us 0.89. And that when we square that value is going to be 0.79 to two significant, sorry, to two decimal places. Now, we're just going to repeat the process for our remaining values, OK? So if we go to the next one. And let me just get my numbers here. So when Y is 35, sorry, when X is 4, OK, then Y hot is going to be 34.61. OK, I know our difference is going to be 0.39 squared, that's 0.15. When X is 5, Y hat is 38.0.11, no, our difference is 0.11 because our Y value is 38, and thus that is gonna be 0.01, OK. Or rather, that should be uh negative. -0.11. OK. Uh, when X is 6. Then our Yha value is gonna be 41.61 and Y minus Y would be negative 0.61 squared, that would be 0.37. When X is 7, Y is 45.11, then we'll have -1.11. And 1.23, uh, when it's 8, Y hot is 48.61. OK. Uh, then we have negative 0.61 square 0.37. Uh, when it's 9, Y height is 52.11. Uh, then our difference is going to be negative 0.11 between it and Y, which means that squared that will be 0.01. Then we're gonna have an X is 10, Y is 55.61. The difference is 0.39 squared 0.15. And finally, when X is 11, Y hat is gonna be 59.11. Difference is going to be 0.89 squared, that's 0.79. So now we have the squares of all our residual values, and when we sum them up, then the sum of our residuals will be 3.87. So this is the value that we're going to plug in to get our standard error of the estimate. So this means then that our standard error of the estimate is gonna be the square root of the sum of the squares of our residuals 3.87 divided by N which is 9 minus 2, and that is approximately equal to 0.74. Thus, the standard error of the estimate is 0.74, which is answer choice C. Thanks a lot for watching everyone. I hope this video helped.

[DATA] The following data represent the height (inches) of boys between the ages of 2 and 10 years.

b. Compute the standard error of the estimate, Sₑ.

Key Concepts

Standard Error of the Estimate (Sₑ)

Linear Regression and Residuals

Sum of Squared Residuals (SSR)

Watch next