Installation of certain hardware takes a random amount of time. The installation times form a normally distributed distribution with a standard deviation 5 minutes and a mean of 45 minutes. A computer technician installs the hardware on 31 different computers. You are interested to find the probability that the mean installation time for the 31 computers is less than 43 minutes. What is the probability that the mean installation time for 31 computers is less than 43 minutes.

To solve this question, the normal distribution and the central limit theorem are used.

Doing this, there is an 0.0129 = 1.29% probability that the mean installation time for 31 computers is less than 43 minutes.

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First, these concepts are presented, then we identify mean, standard deviation and sample size, and then, we find the desired probability.

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Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean and standard deviation , the z-score of a measure X is given by:

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 p-value, we get the probability that the value of the measure is greater than X.

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Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

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Mean of 45, standard deviation of 5:

This means that

Sample size of 31:

This means that

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Probability the sample mean is less than 43:

This is the p-value of Z when X = 43, so:

By the Central Limit Theorem

has a p-value of 0.0129.

Thus, 0.0129 = 1.29% probability that the mean installation time for 31 computers is less than 43 minutes.

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