Question

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

Short answer

The probability that the job will take less than 40 minutes to complete is approximately 0.1587.

Step-by-step solution

Calculate The Total Mean

The total mean is the sum of the means of the individual steps. So, it is \(17 + 13 + 13 = 43\) minutes.

Calculate The Total Standard Deviation

The total standard deviation is the square root of the sum of the squares of the individual standard deviations. So, it is the square root of \(2^2 + 1^2 + 2^2 = 9\), which is 3 minutes.

Standardize The Desired Completion Time

The desired completion time (40 minutes) needs to be standardized using the following formula for z-score: \(z = (x - \mu) / \sigma\), where x represents the score to be standardized, \mu is the mean, and \sigma is the standard deviation. So, \(z = (40 - 43) / 3 = -1\). This step transforms the problem into finding the probability that a normally distributed random variable is less than -1.

Find The Corresponding Probability

Looking at the z-table for normally distributed random variables, the probability that the value is less than -1 is approximately 0.1587.

Key concepts

Normal Distribution

The normal distribution is a fundamental concept in statistics often referred to as a bell curve due to its distinctive shape. When a dataset follows a normal distribution, most observations are clustered around the central peak—indicating the mean (or average)—and probabilities for the values can be easily calculated using various tools, such as z-tables.

The bell curve is perfectly symmetrical about the mean, showing that data points are as likely to fall on either side of the mean. The standard deviation defines the curve's width—it tells us how spread out the data points are. If the standard deviation is small, the curve will be narrow; if it's large, the curve will be wide, indicating more variability in the dataset.

In our exercise, the steps of completing a job are assumed to follow a normal distribution, which allows us to calculate the probability of the job being completed in a certain timeframe using the properties of the normal curve.

Standard Deviation

Standard deviation is a measurement of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean, while a high standard deviation indicates that the values are spread out over a broader range.

Mathematically, the standard deviation is the square root of the variance, which itself is the average of the squared differences from the mean. In practical terms, the standard deviation can provide insights into the reliability and consistency of processes or the expected variability in test results, like the time taken to complete steps in our job-related example.

Knowing the standard deviations and means of the individual steps allows us to calculate the total standard deviation when the steps are independent, which is a critical step in estimating the overall variation in job completion time.

Mean Calculation

The mean, commonly known as the average, is a central measure of location for a dataset. It's calculated by adding up all the values and dividing by the total number of values.

In many practical situations, mean calculation helps us to find the central tendency of a data set, which gives us a general idea of where most values lie. For instance, in the job completion scenario, the mean of the individual steps offers an estimate of the expected time for each step. However, to gauge the total expected time for all steps, we simply add the means of each step (provided the steps are independent), resulting in a total mean time. This aggregate mean will be used as a cornerstone for further probability calculations.

Z-score

A z-score, also known as a standard score, is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point's score is identical to the mean score.

A positive z-score says the data point is above the mean, while a negative z-score says it's below the mean. This metric is particularly useful for comparing individual data points across different normal distributions and for calculating probabilities.

In our exercise, we use the z-score to transform the job completion time into a standard form that can be compared against a standard normal distribution. This allows us to look up the probability of completing the job within a given time frame using standard statistical tables or software.