How well does the number of beers a student drinks predict his or her blood alco

Question

How 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 12-ounce cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC). The data is recorded in the file “bac.xls”. The students were equally divided between man 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. Now, you are going to explore the relationship between the number of beers and BAC though a simple linear regression model in this activity by answering the following questions.

Part 1: Describe the shape, direction, and strength of the pattern you see from the scatter plot:

There is a _____ (straight line/curved) _____ (positive/negative) _____ (weak/strong) relationship between beers and BAC.

Part 2. Fit a least squares regression line.

Does the plot show that the simple linear regression model is a good fit? Consider all the four model assumptions.

Part 3:

Simple linear regression results:
Dependent Variable: Car Price ($100)
Independent Variable: Car Age (years)
Car Price ($100) = 172.06965 - 11.631841 Car Age (years)
Sample size: 10
R (correlation coefficient) = -0.86054085
R-sq = 0.74053055
Estimate of error standard deviation: 10.913728

Parameter estimates:

Analysis of variance table for regression model:

a. Report the equation of the least squares regression line for predicting blood alcohol from number of beers from the “Simple linear regression results”. Round to three decimal places.

BAC = 0.013 + 0.018 Beers

y-intercept = ______________. Interpret its meaning in the context:

slope = _____________. Interpret its meaning in the context:

b. Report SSE from “Analysis of variance table for regression model” table.

SSE = 0.0058496549

Report s as the “Estimated error standard deviation”from the “Simple linear regression results”:

s = 0.020440951

c. Test if there is evidence that drinking more beers increases blood alcohol on the average in the population of all students.

State the hypotheses H0: __________________ Ha: _____________________

Report the value of test statistic, df, and p-value from the “Parameter estimates” table. Notice that the default of Ha is two-sided. You need to convert it for one-sided Ha if necessary.

t=______________

df=______________

p-value=______________

Decision at 0.05 level of significance, reject H0 or not? _____________

Conclusion and interpretation in the context:

d. Report and interpret the coefficient of correlation and the coefficient of determination from “Simple linear regression results”.

Coefficient of correlation r=________. Interpret the direction (positive or negative) and strength (week, moderately strong, or strong) according to its value.

Coefficient of determination R2=_____________ (R-sq in the output). Interpret its meaning in the context:

Part 4. Use the least squares line for prediction.

1. Steve thinks he can drive legally 30 minutes after he drinks 5 beers. Predict Steve’s BAC after 5 beers by a point estimate and by a 95% confidence interval using StatCrunch following the instructions below.

a. In the output window, click “option” botton in the top left corner, and choose “Edit”, and you will go back to the “Simple Linear Regression” setting window.

b. Under “Prediction of Y”, type in “X-value(s)” for Steve.

Predicted values:

Report:

Predicted BAC after 5 beers= ________________

95% prediction interval of Steve’s BAC after 5 beers _______________

The legal limit of BAC is 0.08. Do you think the predicted BAC of Steve’s within the legal limit after 5 beers?

Parameter Estimate Std. Err. Alternative DF T-Stat P-value Intercept 172.06965 11.950432 ? 0 8 14.398613 <0.0001 Slope -11.631841 2.4343055 ? 0 8 -4.7782995 0.0014 BAC 0.15 0.1 0.05 4 8 Beers

Explanation / Answer

Part 1: Describe the shape, direction, and strength of the pattern you see from the scatter plot:

There is a straight line (straight line/curved) positive (positive/negative) strong (weak/strong) relationship between beers and BAC.

Part 2: No, the plot does not show a simple linear regression model is a good fit. A good model has equal number of points on both sides while the given model does not.

Part 3:

a. Report the equation of the least squares regression line for predicting blood alcohol from number of beers from the “Simple linear regression results”. Round to three decimal places.

BAC = 0.013 + 0.018 Beers

y-intercept = 0.013. Interpret its meaning in the context: If the student consumed 0 beer bottles, his BAC would be 0.013

slope = 0.018. Interpret its meaning in the context: Slope represents the rate of change in BAC as No of Beers changes.

c. Test if there is evidence that drinking more beers increases blood alcohol on the average in the population of all students.

State the hypotheses H0: There is no effect of beer on blood alcohol level Ha: _ There is effect of beer on blood alcohol level

t= 0.018/0.020440951 = 0.880585

df=10

p-value=0.2007

Decision at 0.05 level of significance, reject H0 or not? Fail to Reject H0

Conclusion and interpretation in the context: At 95% we fail to conclude if beer consumption increases BAC.

d. Report and interpret the coefficient of correlation and the coefficient of determination from “Simple linear regression results”.

Coefficient of correlation and Coefficient of determination R2 cannot be found as the SSBetween Group is not given. The given information is from a different example.

Part 4:

Predicted BAC after 5 beers = 0.013 + 0.018 * 5 = 0.103

Steve’s BAC is not within the legal limit after 5 beers.

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