# Linear regression T-test

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Linear regression T-test

Crying and IQ Infants who cry easily may be more stimulated than others. This may be a sign of higher IQ. Researchers recorded the crying of 38 infants 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.

How good does this line fit the data?
What are some things that determine how good the line fits the data? R^2 The residual plot. Any obvious curvature in the data?

Conditions/Assumptions/requirements
S---Sample, Identify it! N--Normally distributed residuals(no outliers) A--and I----each piece of data is independent. L----Linear Relationship S---Scattered Residual Plot.

If a baby doesn’t cry, what is their predicted IQ?
If a baby doesn’t cry their predicted IQ is 91.3

Explain the meaning of the slope in context of the problem.
For every unit increase in crycount we can expect an average increase of 1.49 in predicted IQ.

Explain the meaning of the S? Interpret in context of the problem.
S is the standard deviation of the residuals. The avg difference between actual IQ and predicted IQ is about 17.5.

What is the meaning of the correlation coefficient in terms of this problem? What is its value?
R = .445….There is a weak positive linear association between infant crycount and predicted IQ at age 3.

What is the meaning of coefficient of determination in this problem
What is the meaning of coefficient of determination in this problem? What is its value? 20.7% of the variation in predicted IQ can be explained by the approximate linear relationship with infant crycount.

What is the standard deviation of the slope in this problem
What is the standard deviation of the slope in this problem? What does it mean? Seb = .4870……With different samples of this size we can expect a difference in slope to vary as much as

Run the Test Ho: B = 0 vs Ha: B > 0, where B is the the true slope of the relationship between infants crying and later IQ. Assumptions/Conditions………..SNAILS We have a random sample of 38 infants cry counts and IQ measured later at 3 years old. The residuals can be considered Normal because of the CLT. IQ scores can be considered independent and all infants from the selected population is more than 10x the sample. We have a linear association between crying and IQ. The residuals are randomly scattered about the regression line.

Linear Regression T – Test
P-value = .004 This p-value is low enough to reject Ho at any level of alpha. This is strong evidence that there may be a positive linear relationship between infants cry counts and later IQ score.

CI for the slope: Using the data from the crying problem, give a 90% confidence interval for the slope of the problem. Formula is on next slide…..degrees of freedom is N-2 since any two points will make a line, so we discard 2.

Linear Regression T Interval
Conditions have been met. (0.6709, ) We are 90% confident that mean IQ increases is between and point for each additional peak in crying. Since 0 is not included in this interval we have strong evidence that there may be a positive linear relationship between infant cry counts and later IQ.

Blood alcohol content and Beer
16 student volunteers at Ohio State University drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their BAC.

Beers & BAC Student 1 2 3 4 5 6 7 8 Beers 9 BAC 0.10 0.03 0.19 0.12
0.04 0.095 0.07 0.06 10 11 12 13 14 15 16 0.02 0.05 0.085 0.09 0.01 Is there a positive relationship between the number of beers consumed and blood alcohol content(BAC)? Run a Linear Regression T Test…..follow the PHANTOMS

Hypothesis Ho: B = 0 The number of beers has no effect on BAC
Ha: B > 0 The number of beers has a positive effect on BAC

Assumptions/Conditions
We have an independent sample of 16 student volunteers. Residuals are normally distributed and scattered around the regression line(check/show boxplot) We have a linear relationship

Name the Test, Test Statistic & P-value
Linear Regression T – Test t = 7.48 P-value = 0

Make a Decision, Sentence in Context
This p-value is low enough to reject Ho at any level This is very strong evidence to suggest that an increase in the number of beers may increase BAC. Do a 95% CI for the slope of the relationship between BAC and beers.

(.01281, ) Since 0 is not included in our interval we have evidence to suggest that there may be a positive linear relationship between beers consumed and BAC.

If you have drank no beers, what is you predicted BAC
If you have drank no beers, what is you predicted BAC? Is this reasonable? Interpret the slope in context. Interpret the correlation coefficient for this question? Interpret the coefficient of determination for this question? What does the S mean? Interpret this in context.

If no beers are consumed we can expect a BAC of -. 0127
If no beers are consumed we can expect a BAC of Since this is less than zero it is not reasonable. For every beer consumed we can expect on average an increase in BAC of R = .8944….There is a strong positive linear association between beers consumed and predicted BAC. 80% of the variation in BAC can be explained by the approximate linear relationship with beers consumed. S is the standard deviation of the residuals…The average difference between the actual BAC and predicted BAC is

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