Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately 4 hours each to do with a population standard deviation of 1.2 hours. Let X be the random variable representing the time it takes her to complete one review. Assume X is normally distributed. Let be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews. Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

And we can use the z score formula given by:

And using the z score we have:

And we can use the normal standard distribution table or excel in order to find the probabilities and we got:

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 time it takes her to complete one review, and for this case we know the distribution for X is given by:

Where and

And we select a sample size of n =16. Since the dsitribution for X is normal then the distribution for the sample mean is also normal and given by:

For this case we want to find this probability:

And we can use the z score formula given by:

And using the z score we have:

And we can use the normal standard distribution table or excel in order to find the probabilities and we got:

The graph illustrating the problem is on the figure attached.

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step-by-step explanation: