The monthly expenditures on food by single adults living in one neighborhood of Los Angeles are normally distributed with a mean of $370 and a standard deviation of $100. Determine the percentage of samples of size 25 that will have mean monthly expenditures on food within $36 of the population mean expenditure of $370

And we can use the following z score formula:

And replacing we got:

And we want thi probability:

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 expenditures 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 we can use the following z score formula:

And replacing we got:

And we want thi probability:

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