A pediatrician wants to determine the relation that exists...

A pediatrician wants to determine the relation that exists between a child's height, x, and head circumference, y. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the accompanying data. Complete parts (a) through (g) below. Height (inches), x Head Circumference (inches), y 28 17.7 24.25 17.3 25.5 17.2 25.75 17.7 24.5 17.0 28.25 17.8 26.75 17.5 27.25 17.5 26.75 17.5 26.75 17.7 27.5 17.7 (a) Find the least-squares regression line treating height as the explanatory variable and head circumference as the response variable. (b) Interpret the slope and y-intercept, if appropriate. (c) Use the regression equation to predict the head circumference of a child who is 24.5 inches tall. (d) Compute the residual based on the observed head circumference of the 24 .5 -inch-tall child in the table. Is the head circumference of this child above or below the value predicted by the regression model? (e) Draw the least-squares regression line on the scatter diagram of the data and label the residual from part (d). (f) Notice that two children are 26.75 inches tall. One has a head circumference of 17.5 inches; the other has a head circumference of 17.7 inches. How can this be? (g) Would it be reasonable to use the least-squares regression line to predict the head circumference of a child who was 32 inches tall? Why?

Transcript

00:02 All right, so we have a pediatrician who wants to determine the relation that exists between a child's height and its head circumference.

00:10 So here's the height in inches x and the head circumference y in inches as well.

00:18 And the first thing we're going to do is find the at least squares regression line, treating height as the explanatory variable.

00:27 So we're looking for y -hat equals beta -not.

00:32 Plus beta 1.

00:35 You might see this as lowercase b as well.

00:40 Same thing.

00:42 Intercept term, slope term.

00:44 And to do this, we are going to need some statistics of these data.

00:51 So to find the slope coefficient, b1, we need the correlation coefficient times the standard deviation of y is divided by the standard deviation of x.

01:00 And then the beta not, the intercept term, is equal to the mean of the y's minus beta 1 times the mean of the x's.

01:15 So we use the spreadsheet to do all these calculations for us for the mean.

01:23 I'm going to use the average function where you put in the list and it gives you the mean of the data.

01:40 And then for the sample standard deviation, you do std -ev -s to denote the sample.

01:49 And then for the correlation coefficient, you use corell.

01:53 You put in your x's and your y, your x list, and your y list, and out pops the correlation coefficient.

02:01 And so this is what we get.

02:03 Again, let your spreadsheet do the work for you.

02:05 So here are the means and the sample standard evasions of the x's here and the y's.

02:09 Coilation coefficient is this.

02:11 0 .8 and the slope coefficient and the intercept coefficient.

02:17 So our line is then y -hat equals the intercept term is 13 .5.

02:30 The slope term going to three decimals, 0 .151.

02:36 X.

02:37 And there we go.

02:40 And for part b, we're going to interpret these values.

02:44 What is this slope and this intercept term mean? so the intercept term is the head circumference if the height is zero.

02:59 If x is zero, we're going to get the function we'll give us 13 .5.

03:04 But you can't have a height of zero and have a head circumference measurements.

03:09 So it doesn't really make sense in the context.

03:12 This is just a term that anchors our line, if you will.

03:16 And the slope here, 0 .151, this is for every inch, for each inch increase, the head circumference increases by this much.

03:35 So you can think about it as over one, because that's what slope is.

03:41 It's a ratio of the, you know, rise over run going back to algebra class.

03:45 So it's 0 .151 inches in head circumference with a hc for every one inch in height.

04:02 There we go.

04:06 All right.

04:06 Now the next thing we were tasked to do is use this to predict a height or the head circumference of a child with a height of 24 .5 inches.

04:19 So we have this lovely formula.

04:22 Let's use it.

04:24 So y hat is 13 .5, 0 .15.

04:30 And i rounded here.

04:35 0 .5.

04:41 And that gives us a value of 17 .195.

04:46 Great...

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