Question

A certain job is completed in three steps in series. The means and standard deviations for the steps are (in minutes) \begin{tabular}{ccc} \hline Step & Mean & Standard Deviation \\ \hline 1 & 17 & 2 \\ 2 & 13 & 1 \\ 3 & 13 & 2 \\ \hline \end{tabular} Assuming independent steps and normal distributions, compute the probability that the job takes less than 40 minutes to complete.

Short answer

The probability that the job is completed in less than 40 minutes is 0.1587

Step-by-step solution

Find the Total Mean

The total mean time for the job is equal to the sum of the means for each step, which is \(17 + 13 + 13 = 43 \) minutes.

Find the Total Variance

The total variance is found by summing the squares of each standard deviation since variance is the square of the standard deviation. So, the total variance is \(2^2 + 1^2 + 2^2 = 9\).

Find the Total Standard Deviation

The total standard deviation is the square root of the total variance, which is \(\sqrt{9} = 3\) minutes.

Standardize the Desired Time

Use the Z-distribution transformation to standardize the time of 40 minutes. A Z value is computed by subtracting the mean and dividing by the standard deviation. So, the Z value is \((40-43) / 3 = -1\).

Find the Probability

We want to find the probability that time is less than 40 minutes which corresponds to Z < -1, referring to a standard normal distribution table, we find that this probability is 0.1587.