Question

Question 1

Given a normal distribution with µ =15 and σ =5, what is the probability that

1. 5% of the values are less than what X values?
2. Between what two X values (symmetrically distributed around the mean) are 95 % of the values?

INSTRUCTIONS: Show all your work as to how you have reached your answer. Please don’t simply state the results. Show graphs where necessary.

Answer 1

Given = 15 and = 5

To find the probability, we need to find the z scores.

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(a) Here we are being asked for the 5th percentile. therefore P(X < x) = 0.05

The z score at 0.05 is -1.645

Therefore -1.645 = (X - 15) / 5

Solving for X

X = (-1.645 * 5) + 15

X = -8.225 + 15

X = 6.78

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(b) 95% of all calls centered about the mean.

This means that the remaining 5% will be distributed equally to the right and the left = 0.05/2 = 0.025

So the lower p value = 0.025 and the Upper p value = 1 - 0.025 = 0.975

The Z score at p = 0.025 and 0.975 are -1.96 and +1.96 respectively.

The Lower value: (X - 15) / 5 = -1.96. Solving for X, X = (-1.96 * 5) + 15 = 5.2  

The Upper value: (X - 15) / 5 = +1.96. Solving for X, X = (+1.96 * 5) + 15 = 24.8

Therefore 95% will be distributed about 5.2 < X < 24..

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