10) The amount spent per month by employees for parking at a municipal lot is normally distributed with a mean of $60 and a standard deviation of $5. a) About what percentage of the observations lie between $55 and $65? b) About what percentage of the observations lie between $50 and $70? a) About what percentage of the observations lie between $45 and $75?

a)

And we can find this probability with this difference:

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

And that represent 68.2%

b)

And we can find this probability with this difference:

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

And that represent 95.42%

c)

And we can find this probability with this difference:

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

And that represent 97.73%

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".  

Part a

Let X the random variable that represent the amount spent per month by employees 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 with this difference:

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

And that represent 68.2%

Part b

And we can find this probability with this difference:

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

And that represent 95.42%

Part c

And we can find this probability with this difference:

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

And that represent 97.73%

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