1. A manufacturer of a printer determines that the mean number of days before a cartridge runs out of ink is 75 days, with a standard deviation of 6 days. Assuming a normal distribution, what is the probability that the number of days will be less than 67.5 days?

And we can find this probability using the normal standard table or excel:

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the number of days before cartridge runs out of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the normal standard table or excel:

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step-by-step explanation: