Exercise 11.1

1.

(From Bhattacharyya and Johnson (1977), Statistical Concepts and Methods, New York: Wiley). A study was performed to investigate the relationship between speed and stopping distance for an automobile. 10 cars were selected (same year, model, etc.). Each was driven at preassigned speed and when the driver attained that speed the he applied the brakes. The distance to a complete stop was then measured. The data are:

      Speed (X)   :  20   20   30   30   30   40    40    50    50    60
      Distance (Y): 16.3 26.7 39.2 63.5 51.3 98.4  65.7  104.1 155.6  217.2

(a)

Assuming this was a designed experiment what other variables besides car were controlled?

(b)

Scatter plot this data (Y versus X). Comment on the plot. Does it look linear?

(c)

Regardless of your discussion in the last part, use the regression module to fit the model. Predict the stopping distance for an initial speed of 35. Predict the stopping distance for an initial speed of 55.

(d)

Use your predictions in the last part to plot your fit on the scatter plot. Comment? Interpret the estimate of slope.

(e)

Obtain a confidence interval for the slope parameter. What does it mean in terms of the problem? Use it to test H₀ . Conclude in terms of the problem.

(f)

Determine the fit and the residual for the response 98.4 at x = 40.

(g)

Next obtain the residual plot. Does the observation (40, 98.4) seem to be an outlier? Is the scatter random? See the next problem for the answer.

2.

Here is the residual plot for the last problem:

       
                -
              25+                                                       *
                -   *
        Ehat    -
                -   *            *                         *
                -
               0+                *            *
                -
                -
                -                *
                -
             -25+
                -
                -                             *
                -                                          *
                -
             -50+
                  +---------+---------+---------+---------+---------+------ Yhat
                  0        35        70       105       140       175

It is not a random scatter. Sometimes a simple transformation will help. Consider the square root of the stopping distances. These are given by:

      Speed (X)   :  20    20   30    30    30    40     40    50    50    60
      SqrtDistance  4.03  5.16  6.26  7.96  7.16  9.91  8.10 10.20 12.47 14.73

Repeat the last problem using these responses. Notice interpretation changes. As you will see, the residual plot improves considerably but there are still problems with it.

3.

(From Vardeman (1994), Statistics for Engineering Problem Solving, Boston: PWS.) A study was performed to investigate the relationship between the carburetor jetting size and the time of a Camaro for a quarter-mile run. The data are:

             Jet Size     76      68      70      72      74     76
             Time        15.08   14.60   14.50   14.53   14.79   15.02

(a)

Assuming this was a designed experiment what other variables besides car model were controlled?

(b)

Scatter plot this data. Comment on the plot. Does it look linear?

(c)

Regardless of your discussion in the last part, use the regression module to fit the model. Predict the time for a jet size of 76. Predict the time for a jet size of 68.

(d)

Use your predictions in the last part to plot your fit on the scatter plot. Comment? Interpret the estimate of slope.

(e)

Obtain a confidence interval for the slope parameter. What does it mean in terms of the problem? Use it to test H₀. Conclude in terms of the problem.