6.2 Reading a Normal Curve Table

Find the area under the normal curve to the left of z=2.45 To read a normal table, we will need to split the z-value into two parts:

the first part will be the number out to the tenths place and the second part will be the number in the hundredths place.

P(z<1.37) is the probability that

z is less than 1.37.

"Standard Normal Table: Area negative Infinity to z" =

z-values rounded to two decimal places The first decimal place of the z-value is listed down the left-hand column, with the second decimal place across the top row. Where the appropriate row and column intersect, we find the amount of area under the standard normal curve to the left of that particular z-value.

Find the area under the normal curve between z=−1.5 and z=2.65.

First, look up the area to the left of z=−1.5 (using the table " to negative z", which gives you 0.0668. Second, look up the area to the left of z=2.65 (using the table "negative infinity to z", which gives you 0.9960. Last, subtract 0.9960 − 0.0668 = 0.9292. Then the area between our two z-values is 0.9292.

the area to the left of a specific value, x , of the random variable is equal to

P(X<x) (Notice that the inequality symbol points in the direction of the area!)

the area to the right of x equals

P(X>x) (Notice that the inequality symbol points in the direction of the area!)

area and probability are related

Suppose the mean score on Test 1 in your statistics class was 75 with a standard deviation of 5 and the distribution of test scores was normal. What is the probability that a randomly selected test score, such as yours, was better than 80? Look at the normal distribution of test scores shown in the figure: the standard deviation is 5, a score of 80 is one standard deviation above the mean the area under the normal curve above one standard deviation above the mean is equal to 0.1587, so the probability that you scored an 80 or better is 0.1587 or 15.87%.

P(X) stands for the probability that

X will occur

we do not talk about finding the probability that x is a specific value because of the fact that the normal curve is

a continuous probability distribution. Instead, refer to the probability that x is within a range of values. choosing to include the endpoint of our range does not change the value of the probability. So, we can say that P(X<x) = P(X≤x).

The normal curve table, "Standard Normal Table: Area −Infinity to z", only gives the area to the

left of a given z-value we can use the table, along with the properties of the standard normal distribution, to find other areas as well

Method 2 −Find the area under the normal curve to the right of z=2.45.

look up z=−2.45 on the table "negative infinity to -z" which also gives us 0.0071.

Remember that the total area under the standard normal curve is 1. So, if the table gives us the area to the left of z, then

subtracting that area from 1 gives us the area to the right of z.

Method 1 − Find the area under the normal curve to the right of z=2.45.

the area to the left of z=2.45 is 0.9929. So, the area to the right of z=2.45 is 1−0.9929=0.0071

the normal curve is symmetric about

the mean In terms of area under the curve, this means that the area to the right of z is equal to the area to the left of −z.

the area under any part of the normal curve is equal to

the probability of the random variable falling within that region

Remember that probability is the same as area

under the curve. Therefore, P(z>2.36) would be the area under curve to the right of z=2.36 Notice that the inequality sign points the same direction as the area under the curve!

To find the area between two values of z

use the table to look up the area to the left of each z-value and then subtract the smaller area from the larger area.

To find the area to the right of z, instead of looking up the area to the left of z and subtracting that area from 1, you can simply look up

−z.