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 s = 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 need to calculate the area under the normal curve between these two values.
1. Standardize the values: We need to convert the values 36 and 40 into z-scores, which measure the number of standard deviations a data point is from the mean.
z1 = (36 - µ) / s
z2 = (40 - µ) / s
2. Look up the z-scores: We need to find the area between these z-scores under the standard normal curve. We can use a standard normal distribution table or a z-score calculator to find the probabilities associated with these z-scores.
P(z1 < Z < z2)
3. Calculate the probability: Subtract the area to the left of z1 from the area to the left of z2 to find the probability between these two z-scores.
P(z1 < Z < z2) = P(Z < z2) - P(Z < z1)
Using the z-scores formula:
z1 = (36 - 38) / 5 = -0.4
z2 = (40 - 38) / 5 = 0.4
Using a standard normal distribution table or calculator, we find:
P(Z < -0.4) ≈ 0.3446
P(Z < 0.4) ≈ 0.6554
Finally, we calculate the probability:
P(z1 < Z < z2) = P(Z < 0.4) - P(Z < -0.4) = 0.6554 - 0.3446 = 0.31
Therefore, the probability that John's drive to work will take between 36 and 40 minutes is approximately 0.31, or 31%.