Check the model. Recall from Exercise 1 that the mean of the 100 car speeds in Exercise 34 was 23.84 mph, with a standard deviation of 3.56 mph.
a) Using a Normal model, what values should border the middle 95% of all car speeds?
b) Here are some summary statistics.
| Percentile |
|
Speed |
| 100% |
Max |
34.06 |
| 97.50% |
|
30.976 |
| 90.00% |
|
28.978 |
| 75.00% |
Q3 |
25.785 |
| 50.00% |
Median |
23.525 |
| 25.00% |
Q1 |
21.547 |
| 10.00% |
|
19.163 |
| 2.50% |
|
16.638 |
| 0.00% |
Min |
16.27 |
From your answer in part a, how well does the model do in predicting those percentiles? Are you surprised? Explain.
Exercise 1:
Car speeds. John Beale of Stanford, CA, recorded the speeds of cars driving past his house, where the speed limit read 20 mph. The mean of 100 readings was 23.84 mph, with a standard deviation of 3.56 mph. (He actually recorded every car for a two-month period. These are 100 representative readings.)
a) How many standard deviations from the mean would a car going under the speed limit be?
b) Which would be more unusual, a car traveling 34 mph or one going 10 mph?
Exercise 2:
Trees, part II. Later on, the forester in Exercise 3 shows you a histogram of the tree diameters he used in analyzing the woods that was for sale. Do you think he was justified in using a Normal model? Explain, citing some specific concerns.
Exercise 3:
Trees. A forester measured 27 of the trees in a large woods that is up for sale. He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches. Suppose that these trees provide an accurate description of the whole forest and that a Normal model applies.
a) Draw the Normal model for tree diameters.
b) What size would you expect the central 95% of all trees to be?
c) About what percent of the trees should be less than an inch in diameter?
d) About what percent of the trees should be between 5.7 and 10.4 inches in diameter?
e) About what percent of the trees should be over 15 inches in diameter?