Normal Distribution Notes
Name ______________________________
The Normal Curve
The graph of the normal distribution depends on two factors:
1. Mean
2. Standard Deviation
The mean of the distribution _______________________________________________________________
The standard deviation ____________________________________________________________________.
_________________________________________________________________________________________________
Probability and the Normal Curve
The normal distribution is a ______________________________.
The total area under the normal curve is equal to ____________.
About 68% of the area under the curve falls within __________standard deviation of
the mean.
About 95% of the area under the curve falls within __________standard deviations of
the mean.
About 99.7% of the area under the curve falls within _________ standard deviations of
the mean.
Collectively, these points are known as the empirical rule. Clearly, given a normal
distribution, most outcomes will be within __________standard deviations of the mean.
z Score
A z-score (aka, a standard score) indicates how many standard deviations an
element is from the mean. A z-score can be calculated from the following formula.
z
X

where z is the ______________, X is the __________________, μ is the population __________,
and σ is the ________________________.
Here is how to interpret z-scores:
A z-score less than 0 represents an element _________ than the mean.
A z-score greater than 0 represents an element _______________ than the mean.
A z-score equal to 0 represents an element ___________ to the mean.
A z-score equal to 1 represents an element that is _______________________________________
A z-score equal to 2, _____________________________________________________________
A z-score equal to -1 represents an element that is _____________________________________
A z-score equal to -2, ____________________________________________________________
-If the number of elements in the set is large, about 68% of the elements have a zscore between ______________
-about 95% have a z-score between ________________
- and about 99% have a z-score between ________________.
Standard Normal Distribution Table
A standard normal distribution table shows a cumulative probability associated
with a particular z-score.



Table rows show the whole number and tenths place of the z-score.
Table columns show the hundredths place.
The cumulative probability (often from minus infinity to the z-score) appears
in the cell of the table.
Look at your tables: To find the cumulative probability of a z-score equal to -1.31,
cross-reference the row of the table containing -1.3 with the column containing
0.01. The table shows that the probability that a standard normal random variable
will be less than -1.31 is _____________; that is, P(Z < -1.31) = ____________
Find P(Z > a). The probability that a standard normal random variable (z) is greater
than a given value (a) is easy to find.
The table shows the P(Z < a). The P(Z > a) = __________________.
Suppose, for example, that we want to know the probability that a z-score will be
greater than 3.00. From the table (see above), we find that P(Z < 3.00) = 0.9987.
Therefore, P(Z > 3.00) = _______________________
Example 1
An average light bulb manufactured by the Acme Corporation lasts
300 days with a standard deviation of 50 days. Assuming that bulb life is normally
distributed, what is the probability that an Acme light bulb will last at most 365
days? Sketch a normal curve for this situation. (Hint: Find z-score first)
Example 2 Molly earned a score of 940 on a national achievement test. The mean
test score was 850 with a standard deviation of 100. What proportion of students
had a higher score than Molly? (Assume that test scores are normally distributed.)
(A) 0.10 (B) 0.18 (C) 0.50 (D) 0.82 (E) 0.90