Midterm I

1. A high school student tested 3 brands of shoulder pads to determine which brand offered the best protection. To test the shoulder pads, a block of clay was placed under the shoulder pad upon a rectangle (the same size as the block) drawn upon a piece of paper. A 2-pound weight was dropped from varying heights (measured in feet) upon the shoulder pad, compressing the clay. All clay that had been displaced beyond the rectangle was removed from the block and weighed (in oz.). The amount of displaced clay is the response variable and the distance from which the weight was dropped is the independent variable. The brand of shoulder pad is a three-level factor.

Height Displacement Brand

1 .8,1.2,1.3 Champion

2 1.3,1.8 Champion

3 2.0,2.4 Champion

4 3.0,2.5 Champion

1 .5,.3 Rawlings

2 1.1,1.5 Rawlings

3 2.4,2.0,2.4 Rawlings

4 2.7,3.3,3.0 Rawlings

1 2.3,2.7 Team

2 3.6,4.3,3.8 Team

3 5.1,5.7 Team

4 6.6,7.3 Team

(a)

Plot the data, identifying the observations by brand. What do you observe?

(b)

Write a model for E(Y) allowing different slopes and intercepts for each brand.

(c)

Fit the model and test whether the slopes for each brand are equal.

(d)

Test whether the lines for Champion and Rawlings are the same. Assuming, they were indistinguishable, what would the fitted regression equation be?

2. A researcher collected growth data on yellowfin tuna. The response is fork length in centimeters (measured from the nose to the tail fork of each tuna) and the independent variable is age in days (measured from growth rings in the tuna's ear bone-the otolith). (Note: This is actually a calibration problem since the researcher would like to predict age from the more-easily measured variable, fork length.)

(a)

Plot the data

(b)

We will be fitting a logistic growth curve to the data but first need to find good start values for and k. Since is the growth limit and is growth at time 0, you can use the plot in (a) to obtain estimates of and . To find a start value for k, choose any point in the plot and record the x and y coordinates. Plug these values into the growth curve equation along with your estimates of and and solve the equation for k. What are your start values?

(c)

Find a solution. Plot the fitted line and the data and construct a residual plot (you may want to do this in Minitab). Comment on the fit. Interpret and .

(d)

The data has an inflection point at /2 if . Is there strong evidence of an inflection point? Is the results of this test consistent with the appearance of the plot in (a)? Explain any discrepancy.

 
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