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**hidaja16** 1- Given standard normal distribution, find

a) P(Z <-2.33)

Mr F says: Pr(Z < -2.33) = Pr(Z > 2.33) = 1 - Pr(Z < 2.33). Now use your Standard Normal Distribution tables.

b) P(Z> 1.99)

Mr F says: Pr(Z > 1.99) = 1 - Pr(Z < 1.99). Now use your Standard Normal Distribution tables.

C) P(-1.25 <Z<1.75)

Mr F says: Pr(-1.25 < Z < 1.75) = Pr(Z < 1.75) - Pr(Z < -1.25). Get Pr(Z < -1.25) in a similar way to how part a) is done. Now use your Standard Normal Distribution tables.

2 Given standard normal distribution, find the z-score that

a) has 33% of the distribution's area to the right

Mr F says: Get the value of z* such that Pr(Z < z*) = 1 - 0.33 = 0.67 by using your Standard Normal Distribution tables in reverse.

b) corresponds to P5 (the 5th percentile);

Mr F says: Get the value of -z* such that Pr(Z < -z*) = 0.05. Note that Pr(Z < -z*) = 0.05 => Pr(Z > z*) = 0.05 => Pr(Z < z*) = 1 - 0.05 = 0.95. Get z* by using your Standard Normal Distribution tables in reverse.

c) has 75% of the distribution's area between -z and z.

Mr F says: Get the value of z* such that Pr(-z* < Z < z*) = 0.75. Note that Pr(-z* < Z < z*) = 0.75 => Pr(Z < -z*) + Pr(Z > z*) = 2Pr(Z > z*) = 0.25 => Pr(Z > z*) = 0.125 => Pr(Z < z*) = 1 - 0.125 = 0.875. Get z* by using your Standard Normal Distribution tables in reverse.

3. The numbers of hours worked by medical residents is approximately normally distributed with a mean of 81.7 hours and a standard deviation of 6.9 hours.

a) What proportion of medical residents work less than 70 hours per week?

Mr F says: Calculate Pr(X < 70) = Pr(Z < z*) where $\displaystyle {\color{red} z^* = \frac{70 - \mu}{\sigma} = \frac{70 - 81.7}{6.9}}$. Use Q1 to guide you here.

b) If 500 medical residents are selected, how many can be expected to work between 75 and 95 hours in a week?

Mr F says: Calculate Pr(75 < X < 95) = Pr(X < 95) - Pr(X < 75). use the help gven in a) and Q1 to guide you in this.

Then the expected value is $\displaystyle {\color{red} 500 \cdot \Pr(75 < X < 95)}$.

c) What is the probability that a medical resident works more than 100 hours per week?

Mr F says: Calculate Pr(X > 100) = 1 - Pr(X < 100) = 1 - Pr(Z < z*) where $\displaystyle {\color{red} z^* = \frac{100 - \mu}{\sigma} = \frac{100 - 81.7}{6.9}}$.

d) if the bottom 10% are excluded, what is the cut-off number of hours worked per week in order to be included?

Mr F says: You need the value of x* such that Pr(X < x*) = 0.1.

Get the value of -z* such that Pr(Z < -z*) = 0.1. Note that Pr(Z < -z*) = 0.1 => Pr(Z > z*) = 0.1 => Pr(Z < z*) = 1 - 0.1 = 0.9. Get z* by using your Standard Normal Distribution tables in reverse.

Then $\displaystyle {\color{red} -z^* = \frac{x^* - \mu}{\sigma} = \frac{x^* - 81.7}{6.9}}$. Now solve for x*.

e) what value of the number of hours worked per week represents the third quartile?

Mr F says: Find the value of x* such that Pr(X < x*) = 0.75. Use the above questions as a guide.