Women have head circumferences that are normally distributed with a mean given by μ = 22.51 in., and a standard deviation given by σ = 0.9 in. a. if a hat company produces women's hats so that they fit head circumferences between 22.1 in. and 23.1 in., what is the probability that a randomly selected woman will be able to fit into one of these hats?

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 head circunferences 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 on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

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