statistics question

Question

Let Z be the standard normal random variable. Calculate the following and draw a picture illustrating each probability.

(i) P(Z < 1.8) = (ii) P(Z > -1.2) = (iii)

Find c such that P(Z > c) = 0.45

Answer 1

Nora,

 

Rather than calculating the probability of each, I used a z-table to find each:

 

I'd expect you have access to one.

 

First, though, remember that on a z-score normal distribution, a z-score of 0 represents the mean of the distribution. That means that half the area (probability) of the distribution lies below the mean and half above.  Another way to write this is:

 

P(z<0) = .5 and:

 

P(z>0) = .5

 

They need to add to 1.0 since both cover the entire area of the distribution.

 

Now, any value of z that is > 0 lies to the right of the mean (0) and any value of z that is < 0 lies to the left of the mean (0).

 

In the first case, P(z<1.8), a z-score of 1.8 will lie to the right of the mean.  Visualizing this, the probability of a z-score being < 1.8 is the area of the normal curve to the left of 1.8.

 

Now, to find that probability, we first know that the P(z<0) = .5

 

So, P(z<1.8) is going to be .5 + whatever area of the curve is to the right of 0 and to the left of 1.8.  This is the area given by the z-table.

 

If I look at the z-table, I find that the value of the area above 0 for a z-score of 1.8 is .464.  So, adding .5 to .464, we get:

 

.964 which is P(z<1.8).  This represents the area of the normal curve to the left of the z-score 1.8.

 

See if you can do part ii the same way (remember that now you're looking for the area of the curve to the RIGHT of the z-score -1.2); you'd have the .5 (everything to the right of the mean [0]) added to the area between -1.2 and 0.

 

For part iii, you're looking for the z-score where the area of the curve to the right of it is equal to 0.45.

 

If you think about this for a second, you can see that the z-score in question will lie to the right of the mean (0) since .5 or half the curve's area is to the right of the mean.  Our z-score's probability is less than .5 so it needs to be on the right of the mean.  In fact, it will lie pretty close to the mean.

 

To find the z-score, you need to find the area between 0 and c.  We know this area since everything to the right of 0 is .5 and everything to the right of c is .45.  As a result, the area between 0 and c is:

 

.5 - .45 = .05  

 

The table will tell us what the z score we're looking for is if we look for .05 in the table.  When you do that, you find that a z-score of .103 has a probability of .0517 which is close (many are not exact).  So:

 

P(z>.013) = 0.45

 

I hope I haven't confused you!

 

This would be easier to understand if I could draw a diagram.

Answer 2

Nora,

 

You need a table for the cumulative distribution of the standard normal variable z. Most statistics books include such a table. Your calculator should also have a function that gives the value of the cumulative distribution given a value of z.

 

P(z < 1.8) can be read from the table or your calculator for z = 1.8 (actually that entry is for z < 1.8).

 

Draw a picture of a bell-shaped curve centered on z = 0 with a standard deviation of 1.0. z = 1.8 appears to the right of the center. You should shade the area under the curve to the left of z = 1.8. If your calculator cannot print such a curve, plot each value of z from -2.0 incrementing by 0.1 until +2.0 from the table or your calculator.

 

P(z > -1.2) can be represented on a similar bell-shaped curve by shading the are under the curve to the right of z = -1.2, which appears to the left of center.

 

These values are the probabilities NOT less than z = 1.2 or P(z > -1.2) = 1 - P(z < -1.2).

 

Finally, look through the table for the (cumulative) probability 0.45. The value of the row in which that value is found and (added to) the value of the column in which that value is found gives you the value for c. This is just the opposite of the procedure for finding a probability in the table given a value for c.

 

Again, you can create that bell-shaped curve centered on z = 0 with a standard deviation of 1.0 and shade the area under the curve to the left of z = c. The shaded are should be a little less than half of the total area under the curve.

 

Michael.