Question

Yoonie is a personnel manager in a large corporation. Each month she must review 25 of the employees. From past experience, she has found that the reviews take her approximately 5 hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let X ¯ be the random variable representing the mean time to complete the 25 reviews. Assume that the 25 reviews represent a random set of reviews.

1. Find the probability that one review will take Yoonie from 4.5 to 5.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability. . P (________ < x < ________) = _______

2. Find the probability that the mean of a month’s reviews will take Yoonie from 4.5 to 5.25 hrs. Sketch the graph, labeling and scaling the horizontal axis.

Shade the region corresponding to the probability. P (________________) = _______

What causes the probabilities for these two to be different?

Answer 1

Q 1

$P(4.5<X<5.25) =P(\frac{4.5-5}{1.2}<Z<\frac{5.25-5}{1.2})$

Q 2)

$P(4.5<\overline{X}<5.25) =P(\frac{4.5-5}{1.2/\sqrt{25}}<Z<\frac{5.25-5}{1.2/\sqrt{25})})$

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The probabilities for these two are different because as the sample size increased, the margin of error decreases.