A study carried out to investigate the distribution of total braking time (reaction time plus accelerator-to-break movement time, in ms) during real driving conditions at 60 km/hr gave the following summary information on the distribution of times ("A Field Study on Braking Responses Dring Driving," Ergonomics, 1995: 1903-1910) : mean = 535, s = 80. Assume the braking times have approximately a bell shaped distribution. A) Approximately what percentage of braking times are in the interval (535, 615) ? B) What value do the top 2.5% of braking times exceed? (show work)

a) Approximately 34% of braking times are in this interval.

b) 695 ms

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 535

Standard deviation = 80

A) Approximately what percentage of braking times are in the interval (535, 615) ?

The empirical rule is symmetric, which means that 50% of the measures are below the mean and 50% are above.

615 = 535 + 80

So 615 is one standard deviation above the mean.

535 is the mean.

Of the measures above the mean (50%), 68% are within 1 standard deviation of the mean.

Then

0.5*0.68 = 0.34

Approximately 34% of braking times are in this interval.

B) What value do the top 2.5% of braking times exceed?

95% of the measures are within 2 standard deviation of the mean. That is, between the bottom 100 - (95/2) = 2.5% and the top 100 + (95/2) = 2.5% are within 2 standard deviations of the mean.

The top 2.5% are more than two standard deviations of the mean.

535 + 2*80 = 695

So the top 2.5% of braking times exceed 695 ms.