Confidence Intervals for Confidence Intervals for the Regression Slopethe Regression Slope

12.1b12.1bTarget Goal: Target Goal: I can perform a I can perform a significance test about the slope β of a significance test about the slope β of a population (true) regression line.population (true) regression line.

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Review Linear RegressionReview Linear Regression

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Height

Wei

ght

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How much would an adult male weigh if

he were 5 feet 4 inches tall?

Weights of men will vary – in other words,

there is a distribution of weights for adult males who are 5 feet 4 inches

tall.

This distribution is normally

distributed.

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We want the standard

deviations of all these normal

distributions to be the same.

Let’s look at the heights and weights of a Let’s look at the heights and weights of a population of adult men.population of adult men.

Are some of these weights more likely

than others?What would this distribution look

like?

What would you expect for other heights?

Where would you expect the population

regression line to be?

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The Sampling Distribution of The Sampling Distribution of b : Old Faithful b : Old Faithful cont.cont.

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6.159: 1.30

1 1.083 20 1 b

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Spread

Let’s return to our earlier exploration of Old Faithful eruptions. For all 222 eruptions in a single month, the population regression line for predicting the interval of time until the

next eruption y from the duration of the previous eruption x is µy = 33.97 + 10.36x.

The standard deviation of responses about this line is given by σ = 6.159. If we take all possible SRSs of 20 eruptions from the population, we get the actual sampling distribution of b.

Shape:Center : (b is an unbiased estimator of β)

In practice, we don’t know σ for the population regression line. So we estimate it with the standard deviation of the residuals, s. Then we estimate the spread of the sampling distribution of b with the standard error of the slope:

1b

x

sSE

s n

Approx. Normalµb = β = 10.36

Performing a Significance Test for the Performing a Significance Test for the SlopeSlope

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Suppose the conditions for inference are met. To test the hypothesis H0 : β = hypothesized value, compute the test statistic

Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis Ha. Use the t distribution with df = n - 2.

Suppose the conditions for inference are met. To test the hypothesis H0 : β = hypothesized value, compute the test statistic

Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis Ha. Use the t distribution with df = n - 2.

0

b

bt

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Typically = 0

Example: Crying and IQExample: Crying and IQIn

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Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants 4 to 10 days old and their later IQ test 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 its 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. A scatterplot and Minitab output for the data from a random sample of 38 infants is below.

Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants?

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State: We want to perform a test ofH0 : β = 0 Ha : β > 0

where β is the true slope of the population regression line relating crying count to IQ score. No significance level was given, so we’ll use α = 0.05.

Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants?

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Plan: If the conditions are met, we will perform a t test for the slope β.

• Linear: The scatterplot suggests a moderately weak positive linear relationship between crying peaks and IQ.

The residual plot shows a random scatter of points about the residual = 0 line.

• Independent:

Later IQ scores of individual infants should be independent.

Due to sampling without replacement, there have to be at least 10(38) = 380 infants in the population from which these children were selected.

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• Normal: The Normal probability plot of the residuals shows a slight curvature, which suggests that the responses may not be Normally distributed about the line at each x-value. With such a large sample size (n = 38), however, the t procedures are robust against departures from Normality.

• Equal variance: The residual plot shows a fairly equal amount of scatter around the horizontal line at 0 for all x-values.

• Random: We are told that these 38 infants were randomly selected.

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Do: With no obvious violations of the conditions, we proceed to inference. The test statistic and P-value can be found in the Minitab output.

Conclude: The P-value, 0.002, is less than our α = 0.05 significance level, so we have enough evidence to reject H0 and conclude that there is a positive linear relationship between intensity of crying and IQ score in the population of infants.

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The Minitab output gives P = 0.004 as the P-value for a two-sided test. The P-value for the one-sided test is half of this,P = 0.002.

Testing the Hypothesis of No Testing the Hypothesis of No Linear RelationshipLinear Relationship

To test whether or not there is a To test whether or not there is a correlation correlation between between two two quantitative variablesquantitative variables, consider the , consider the slope of the regression line.slope of the regression line.

If there is If there is no correlationno correlation, the , the slope slope would be would be zerozero. .

HH00: : ββ = 0 = 0(The mean of y does not change at all as x (The mean of y does not change at all as x

changes.)changes.)

To test this hypothesis, compute To test this hypothesis, compute the the tt statistic and P-value. statistic and P-value.

Note:Note: Regression Regression outputoutput from statistical from statistical

software usually software usually gives gives tt and its and its two-two-sidedsided P-value. P-value.

For a one-sided testFor a one-sided test,, divide the P-divide the P-value in the output by 2.value in the output by 2.

b

bt

SE

Ex: Beer and Blood AlcoholEx: Beer and Blood Alcohol

How well does the number of beers a How well does the number of beers a student drinks predict his or her blood student drinks predict his or her blood alcohol content? alcohol content?

Sixteen Sixteen of age of age college student college student volunteers drank a randomly assigned volunteers drank a randomly assigned number of cans of beer. number of cans of beer. Thirty minutes Thirty minutes later, a police officer measured their later, a police officer measured their blood alcohol content (BAC).blood alcohol content (BAC).

Data revealed:Data revealed:

They noticed a variation in the data They noticed a variation in the data and didn’t believe that the number of and didn’t believe that the number of drinks predicted the BAC well. Here is drinks predicted the BAC well. Here is the scatter plot.the scatter plot.

The solid line is the LSRL.The solid line is the LSRL. The scatterplot shows a The scatterplot shows a clear linear clear linear

relationship.relationship.

Minitab output: Minitab output:

Because r2 = 0.8000, the number of drinks accounts for 80% of the observed variation in BAC.(The students are wrong).

To test hypothesis that the To test hypothesis that the number of beers has no effect on number of beers has no effect on

BAC;BAC;HHoo: : ββ = 0 = 0 There is no correlation between There is no correlation between the number of beers consumed and BAC.the number of beers consumed and BAC.

HHaa: : ββ > 0 > 0 There is a positive correlation There is a positive correlation between the number of beers consumed between the number of beers consumed greater the BAC.greater the BAC.

Examine the P-valueExamine the P-value From the output: From the output: P-value = 0.0000P-value = 0.0000, , The The one sided P-valueone sided P-value is is half half of this so of this so

it is also close to 0.it is also close to 0.

Thus, Thus, we reject Hwe reject Hoo and conclude and conclude that the number of beers that the number of beers does have does have an effect on BAC.an effect on BAC.

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