Question

The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.43. seconds and a standard deviation of 0.12 seconds. What is the reaction time that is exceeded 296 of the time?

Answer 1

Step-by-step explanation:

To find the reaction time that is exceeded 296 out of 1000 times (or 29.6% of the time), we need to find the corresponding z-score and then convert it back to the original reaction time value.

Given:

Mean reaction time (μ) = 0.43 seconds

Standard deviation (σ) = 0.12 seconds

Proportion of times exceeded (p) = 0.296

First, we need to find the z-score corresponding to the given proportion using the standard normal distribution table or a statistical calculator.

The z-score can be calculated using the formula:

z = (x - μ) / σ

where:

x is the reaction time value we want to find

μ is the mean reaction time

σ is the standard deviation

Rearranging the formula, we have:

x = z * σ + μ

Now, let's calculate the z-score:

z = Φ^(-1)(1 - p)

where Φ^(-1) represents the inverse of the cumulative distribution function (CDF) of the standard normal distribution.

Using a standard normal distribution table or a calculator, we find that Φ^(-1)(0.704) ≈ 0.525.

Now, we can calculate the reaction time value (x) exceeded 29.6% of the time:

x = 0.525 * 0.12 + 0.43 ≈ 0.493 seconds

Therefore, the reaction time that is exceeded 296 out of 1000 times (29.6% of the time) is approximately 0.493 seconds.