1. ## Normal Distributions

This is normal distributions and i dont know anything about how to do this or where to even start so any help would help me Thanks \frac{\mathrm{d} }{\mathrm{d} }

The time required to finish a test in normally distributed with a mean of 40 minutes and a standard deviation of 8 minutes. What is the probability that a student chosen at random will finish the test in more than 48 minutes?
A.84%
B.2%
C.34%
D.16%
2.
Suppose you have a normally distributed set of data pertaining to a standardized test. The mean score is 1000 and the standard deviation is 200. What is the z-score of 1600 point score?
A.2.5
B.3.0
C.1.5
D.0.5

2. The z-score is defined as follows:

$\displaystyle z = \frac{x - \mu}{\sigma}$

Where x is the observed mean, $\displaystyle \mu$ is the population mean, and $\displaystyle \sigma$ is the standard deviation. The z number converts the given values into standard units based on the standard deviation. Consider the first question. What is our z-score? The difference of the observed is 8 greater than the mean (= 48 - 40). Thus, the observed mean is one standard deviation from the mean. One thing you should know is that 68% of the population is within one standard deviation of the mean. This means that 100% less 68% (= 32%) of the observations fall outside of one standard deviation from the mean. But we're not concerned with just all observations. We're concerned with those on the positive side: i.e., for those observations $\displaystyle \geq 48$. Since the normal distribution is symmetric, there is an equal number on both sides of one standard deviation from the mean. Therefore, we only need to consider half. Therefore, 16 (= 32/2) percent of the observations lie positively one standard deviation from the mean. In other words, you have a 16% chance of being observed $\displaystyle \geq 48$ given this normal distribution.

Did you understand that? The second question is just a calculation. Interpret it correctly: how many standard deviations is 1600 from the mean of 1000 when $\displaystyle \sigma = 200$?