Question

A large disaster cleaning company estimates that 30 percent of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that it has contracted, find the probability that x (number of jobs finished on time) is within one standard deviation of the mean.

Answer 1

68.26% probability that the number of jobs finished on time is within 1 standard deviation of the mean.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Looking at a random sample of 8 jobs that it has contracted, find the probability that x (number of jobs finished on time) is within one standard deviation of the mean.

Within 1 standard deviation of the mean is from Z = -1 to Z = 1. So this probability is the pvalue of Z = 1 subtracted by the pvalue of Z = -1.

Z = 1 has a pvalue of 0.8413

Z = -1 has a pvalue of 0.1587

So there is a 0.8413 - 0.1587 = 0.6826 = 68.26% probability that the number of jobs finished on time is within 1 standard deviation of the mean.