Chapter 5 Practice Test: Summarizing Bivariate Data
Name ______
The Roman Empire minted coins in both Rome and the Eastern provinces. Historians would like to use metallurgical analysis as one tool to identify the source mints of Roman coins. They analyzed 11 coins known to have been produced by Eastern mints in an attempt to identify a trace element profile, and have identified gold and lead as possible factors in identifying other coins’ source mint. The gold and lead content of the 11 coins, measured as a percentof weight, is given in the table at right. A scatter plot of these data is displayed below.
1. a) What is the equation of the least squares
best fit line for predicting lead content
from gold content?
b) Sketch the best fit line on the scatter
plot.
c) What is the value of the correlation
coefficient? Interpret this value in the
context of this problem.
d) What is the value of the coefficient of determination? Interpret this value in the
context of this problem.
2. Suppose that the coins analyzed in problem 1 are representative of the metallurgical content of mints in the Eastern provinces of the Roman Empire.
a) If a coin from the Eastern provinces is selected at random, and its gold content is
0.30% by weight, what is the predicted lead content?
b) One of the coins used to calculate the regression equations has a gold content of
0.30% and a lead content of 0.17%. Calculate the residual for this coin.
3. When children are discharged from the hospital their parents may still provide substantial care, such as the insertion of a feeding tube. It is difficult for parents to know how far to insert the tube, especially with rapidly growing infants. Health care professionals believe it may be possible for parents to measure their child’s height and by using a regression equation calculate the appropriate insertion length using a regression equation. At a major children’s hospital, children and adolescents’ heights and esophagus lengths were measured and a regression analysis performed. The data from this analysis is summarized below:
Summary statistics from Regression Analysis
Height (cm) and Esophageal Length (cm)
EsophagusLength (E) / Height
(H)
/ 34cm / 124.5cm
s / 3.5cm / 19cm
a) For a child with a height one standard deviation above the mean, what would be the
predicted esophagus length?
b) What proportion of the variability in esophagus length is accounted for by the height
of the children and adolescents?
c) From the information presented above, does it appear that the esophagus length can be accurately predicted from the height of young patients? Provide statistical evidence for your response.
4. Hemorrhagic disease in white-tailed deer is caused by a virus known as EHD. Immunity is given to fawns by transfer of EHD antibodies from the mother. In a study to determine how long the maternal antibodies last, blood samples were taken from a large representative sample of fawns of known age. The mean EDH concentration (E) was determined for the fawns that were 1 week old, the fawns that were two weeks old, and so on. The mean levels of EHD antibody concentration are given in the table below.
After using the data to fit a straight line model, significant curvature was detected in the residual plot. Two nonlinear models were chosen for further analysis, the exponential and the power models. (Common logs were used to perform the transformations.) The computer output for these models is given below, and the residual plots are on the next page.
(Exponential)
(Power)
Residual Plots
a) For the exponential model, calculate the predicted log of the EHD antibody concentration for an age of 5 weeks.
b) Use your calculations from part (a) to predict the EHD antibody concentration for an age of 5 weeks.
c) Generally speaking, which of the two models, power or exponential, is better at
predicting the log of the EHD antibody concentration? Provide statistical justification for your choice based on both the residual plot and the numeric summary above.
d) The researchers would like to use whichever is the best model to predict EHD antibody concentrations for fawns aged up to 24 weeks. Do you feel this would be reasonable? Explain why or why not.
Chapter 5 Test, Form A
Page 1 of 7