INFERENCE FOR REGRESSIONIs a child’s IQ linked to their crying as an infant?

Crying and IQInfants who cry easily may be more easily stimulated than others and this may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants four to ten days old and their later IQ scores. A snap of a rubber band on the sole of the foot caused the infants to cry. The researchers recorded the crying and measured intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test.

The data for this experiment is on the next page (Table 14.1 in our book).

Crying and IQ

Crying IQ Crying IQ Crying IQ Crying IQ

10 87 20 90 17 94 12 94

12 97 16 100 19 103 12 103

9 103 23 103 13 104 14 106

16 106 27 108 18 109 10 109

18 109 15 112 18 112 23 113

15 114 21 114 16 118 9 119

12 119 12 120 19 120 16 124

20 132 15 133 22 135 31 135

16 136 17 141 30 155 22 157

33 159 13 162

Crying and IQ Plot and Interpret

Input the data in your calculator and create a scatterplot

Look for form, direction and strength in the plot

Numerical Summary Calculate a LinReg(a+bx) L1, L2, y Make note of the value of r

Mathematical Model We are interested in predicting the response (IQ)

from the explanatory (crying) Create an LSRL using a and b

Crying and IQSlope and Intercept

The slope of the regression line () is of particular importance. A slope is a rate of change

The true slope says how much higher average IQ is for children with one more peak in their crying measurement.

Because estimates the unknown , we estimate that on the average IQ is about 1.5 points higher for each added crying peak.

Crying and IQSlope and Intercept

Though we need the intercept () to draw the line, it has no statistical meaning in this example.

No child had fewer than 9 crying peaks, so we have no data near .

It is safe to expect that all normal children would cry when snapped by a rubber band, thus we will never observe .

Crying and IQResiduals and Standard Error

The data table shows that the first recorded infant had 10 crying peaks and a later IQ of 87.

The predicted IQ for is

The residual for this observation is

That is, the observed IQ for this infant lies 19.2 points below the least-squares line on the scatterplot.

Crying and IQResiduals and Standard Error

Let L3 = RESID or L3 = L2 – Y1(L1) Verify that the sum of the residuals is 0

The variance about the line is found with…

The standard error about the line is found with …

Crying and IQResiduals and Standard Error

I would suggest letting L4 = L32

What is the variance of our data?

What is the standard error of our data?

Crying and IQConfidence intervals for the regression slope

Remember, we will not be able to find , but we can use to find a range of values in which we can be confident is contained…

The standard error of the least-squares slope is…

Where is the upper critical value from the distribution with degrees of freedom

Crying and IQConfidence intervals for the regression slope

The output below for the crying study is from the regression command in the Minitab software package. Most such packages have similar output.

The first line gives the equation of the lest-squares regression line.

Crying and IQConfidence intervals for the regression slope

Let’s look at the rest of the information given.

Crying and IQConfidence intervals for the regression slope

The hypothesis says that crying has no straight-line relationship with IQ. Our previous work showed that there is a relationship. The analysis above gives us two values that give us

very strong evidence that IQ is correlated with crying.

Beer and Blood AlcoholHow well does the number of beers a student drinks predict his or her blood alcohol content? Sixteen student volunteers at Ohio State university drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC). Here are the data:

Student: 1 2 3 4 5 6 7 8

Beers: 5 2 9 8 3 7 3 5

BAC: 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06

Student: 9 10 11 12 13 14 15 16

Beers: 3 5 4 6 5 7 1 4

BAC: 0.02 0.05 0.07 0.10 0.085 0.09 0.01 0.05

Beer and Blood Alcohol

The students were equally divided between men and women and differed in weight and usual drinking habits. Because of this variation, many students don’t believe that number of drinks predicts blood alcohol well. What do the data say?

Student: 1 2 3 4 5 6 7 8

Beers: 5 2 9 8 3 7 3 5

BAC: 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06

Student: 9 10 11 12 13 14 15 16

Beers: 3 5 4 6 5 7 1 4

BAC: 0.02 0.05 0.07 0.10 0.085 0.09 0.01 0.05

Beer and Blood Alcohol

Let’s input the data into our calculators

Beers will be our explanatory variable

BAC will be our response variable

Student: 1 2 3 4 5 6 7 8

Beers: 5 2 9 8 3 7 3 5

BAC: 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06

Student: 9 10 11 12 13 14 15 16

Beers: 3 5 4 6 5 7 1 4

BAC: 0.02 0.05 0.07 0.10 0.085 0.09 0.01 0.05

Beer and Blood AlcoholH0: number of beers has no effect on BAC

Ha: number of beers increases BAC

Do a linear regression on the data

Create a list of the residuals in L3

Square the list of the residuals in L4

Beer and Blood Alcohol Calculate the value of s

Store this value in S

Beer and Blood Alcohol Calculate

Store this value in E

Beer and Blood Alcohol Why did we find ?

It is needed to find the confidence interval for

is the value from the table for 95% confidence with 14 degrees of freedom

What does this interval tell us (in context)? We are 95% confident that the true slope is between

these two values.

Beer and Blood Alcohol Now, let’s determine if we have statistically

significant evidence that the number of beers effects BAC

We need to find the -ratio

Store this value in

Beer and Blood Alcohol According to the hypotheses, is this a one- or

two-sided test?

Using the -ratio we just found, calculate our p-value

Beer and Blood Alcohol What is our conclusion to this point?

With a p-value so small ( we have very strong statistical evidence that the number of beers a person drinks does elevate their BAC. Specifically, to a 95% level of confidence, for each beer a person consumes, their BAC should increase between 0.013% and 0.023%. However, student number 3’s BAC was 0.04% (0.19 – 0.15) higher than normal. Though the is not extreme for this student, it is possible this value is influential. To verify that our results are not too dependent on this one value, removal of it and recalculating may be necessary.

Beer and Blood Alcohol Remove Student 3 and recalculate

a = 2.481E-5 b = 0.0146 r2 = 0.7684 r = 0.8766 S =

as opposed to 0.077

Predicting Blood AlcoholSteve thinks he can drive legally 30 minutes after he finishes drinking 5 beers. We want to predict