John drives to work each morning and the trip takes an average of µ = 38 minutes. The distribution of driving times is approximately normal with a standard deviation of ó = 5 minutes. For a randomly selected morning, what is the probability that John’s drive to work will take between 36 and 40 minutes?
To find the probability that John's drive to work will take between 36 and 40 minutes, we can use the concept of z-scores and the standard normal distribution.
Step 1: Convert the given times to z-scores.
We know that the average driving time is 38 minutes (µ = 38) and the standard deviation is 5 minutes (σ = 5). To find the z-scores for 36 and 40 minutes, we can use the formula:
z = (x - µ) / σ
For 36 minutes:
z1 = (36 - 38) / 5
z1 = -0.4
For 40 minutes:
z2 = (40 - 38) / 5
z2 = 0.4
Step 2: Use the standard normal distribution table or calculator to find the probability between the z-scores.
The standard normal distribution has a mean of 0 and a standard deviation of 1. We want to find the probability between z1 = -0.4 and z2 = 0.4.
By referring to a standard normal distribution table or using a calculator, we can find that the cumulative probability (area under the curve) from z = -0.4 to z = 0.4 is approximately 0.1554.
Step 3: Interpret the result.
The probability that John's drive to work will take between 36 and 40 minutes is approximately 0.1554, or 15.54%.
Note: The standard normal distribution is an approximation of the driving times distribution being approximately normal. It is assumed that the driving times are normally distributed in this context.