Normal Curve Example 1

Examples
Adapted from the forthcoming CyberStats introductory statistics course.

Example 1 - Speed Limits and Vehicle Speeds

Background:
In 1987, states were allowed to raise the maximum speed limits on their rural interstate highways to 65 miles per hour. Prior to 1987, the national maximum speed limit in the United States was 55 mph, a limit legislated in 1973. How much change in vehicle speeds actually resulted from the increased speed limit? In the late 1980s, the National Highway Traffic Safety Administration (NHTSA) funded a research project to examine the effects of the speed limit on speeds.

The Research Design
The research design was to compare speed data measured during the two years before the 65 mph limit to speed data measured during the two years after the increased limit.

A Question of Interest
What was the change in the percentage of vehicles traveling faster than 70 mph? NHTSA was concerned that a marked increase in this percentage would lead to increases in highway fatalities.

The Data

For the 55 mph years, means and standard deviation were available for a large sampling of interstate locations because states were required to report these statistics to the Federal Highway Administration.
For the 65 mph years, the research group measured speeds at several locations in 13 different states during 1988 and 1989.

Typical Summary Statistics
Means and standard deviations varied somewhat from one site to another. The following table shows typical summary statistics for the two different speed limit periods.

Speed Limit	Mean Speed	St. Dev.
55 mph	61 mph	6 mph
65 mph	66 mph	6 mph

Use of The Normal Curve Model

The normal curve model was a reasonable approximation to the distribution of the data collected in 1988 and 1989. Assuming this was also true in 1985 and 1986, the published means and standard deviations for those years are sufficient information to use the normal curve model for those years.

What Percentage of Vehicles Travel Faster Than 70 mph?

Faster than 70?

For the 55 mph era:

Mean= 61, standard deviation = 6 .
The standardized score for 70 is
or, Z = (70 - 61 ) / 6 = 1.5 .
For Z = 1.5, the percentile rank is 0.9332 , about 93%

Use the calculator below to check this.

The percentage traveling faster than 70 mph is about 100-93 = 7% .

Z Score

Prob<Z

For the 65 mph era:

mean= 66, standard deviation = 6

The standardized score for 70 mph is Z=0.67. Check this.

For Z = 0.67, the percentile rank is about 0.75, or 75% . Use the calculator above to check this.

The percentage traveling faster than 70 mph is about 100 - 75 = 25% .

Summary and Interpretation

In 1985 and 1986, when the maximum speed limit was 55 mph, about 7% of vehicles were traveling faster than 70 mph on rural interstates.

This percentage increased to about 25% in 1988 and 1989, when the speed limit was 65 mph.

The normal curve model was used to make these estimates because only means and standard deviations were available for the 55 mph era.

Now work through Example 2. It will test your understanding of the normal distribution.

Example 1
Example 2