normal distribution that has a mean of 100,

Question

A. For a normal distribution that has a mean of 100, and a standard deviation of 8, determine the Z-score for each of the following X values:
1 B. Use the information in 1 A. to determine the area or probability of the following: p(84 < x < 116),   p(84 < x < 116),   p(x < 84)

Answer 1

I don't know what method you're supposed to be using to do this - it can also be difficult to explain without the use of visuals of some sort.

 

If you're using a normal distribution chart, I know at least 3 ways these can be set up, so it would also depend on how your particular chart(s) works, so I will just attempt to explain the idea behind it.

 

Notice that 84 is 2 standard deviations to the left of the mean, while 116 is 2 standard deviations to the right of the mean.  So you've got the same distance on both sides.  You should have gotten those numbers from the z score in part a.

 

The probability of something being between the mean and 2 standard deviations is .4772.  Since the range goes both directions, you have 2 of those (one left and one right), so that is .9544.  (If you learned the empirical rule, that's where "about 95%" came from.)

 

The first two ranges look the same, so I'm not sure if one of those was meant to include "equal to," but it doesn't make any difference.  The probability of anything equaling something is 0, so "greater than" and "greater than or equal to" become the same thing.

 

Note the third one (< 84) goes to the left, and 84 is already 2 standard deviations to the left of the mean.  So that's the left tail.  You may have a chart that only has tails on it, in which case you can directly look up 2 on the chart. 

 

The other way is to remember that from the mean to 84 to the left is .4772.  Half of the entire distribution is .5.  (All probabilities add up to 1, so the entire area under the curve is 1, so half that is .5.)  If you take .5 - .4772, that gives you that tail that's left.

 

I would highly recommend that you draw the distribution, mark 3 slashes left and 3 right, to visualize the standard deviations, and mark a line at the spot where 84 and 116 are, and then visualize the area these refer to.  Pictures are truly worth it for this material, and it can get difficult to explain without them.