Download Determining Probabilities Under the Normal Curve

Determining Probabilities Under the Normal Curve
So far we have looked at boundaries for shaded regions under the normal curves; they have been exactly
one or two standard deviations away from the mean. However, an individual’s z-score in any kind of
continuous probability distribution can equal a decimal value that is between or beyond these whole zscores. So to work with this, we need to use more ways to find the probabilities within the given range of
scores.
All the values within the z-score table represent the area under the curve between the mean of the
distribution and the given z-scores. Since this is a z-score table as opposed to an actual data table, the
mean of the data is assumed to be zero, while the standard deviation is 1.
Positive Z-scores
Use the z-score table to calculate the probability of the indicated range of z-scores.
(a) p(0<z<1.2)
(b) p(z<1.2)
(c) p(z<1.85)
(d) p(z>0.73)
(e) p(0.5<z<2.0)
(f) p(z=0.33)
Negative Z-scores
Negative scores are below the mean.
Example
p(z<-0.67)
p(z>0.67)
Example
Use the z-score table to calculate the probability of the indicated range of z-scores.
(a) p(-0.4<z<0)
(b) p(z<-0.4)
(c) p(z>-0.43)
(d) p(-0.55<z<1.55)
(e) p(-0.67<z<0.67)
(f) (-1.8<z<0)
Revisiting Z-score Calculations
Example
Determine Michael’s z-score on a physical examination if he scored 70 out 100, where the mean of the
population is 66 with a standard deviation of 8.
Example
Find the mean of a set of data with a standard deviation of 12, given that a value of 74 has a z-score of 1.5.
Raw Data and the Z-score Table
In many situations, data is not presented as a z-score, but as raw values relevant to a given situation. For
instance, the scores on an exam might be normally distributed with a mean of 75 and a standard deviation
of 9. You must convert any raw data into z-scores before you can carry out any probability analyses using
the z-score table.
Example
Find the probabilities for the following X values, given the mean and standard deviation of the data set:
(a) Given μ= 5 and δ = 2, find p(X<4.5)
(b) Given μ= 2 and δ = 0.2, find p(X>1.5)
(c) Given μ= 50 and δ = 10, find p(X<55)
Real-world Applications of the Normal
Distribution
Example
The life span of a particular species of turtles is normally distributed with a mean of 180 years and a
standard deviation of 40 years.
(a) What is the probability that a randomly chosen turtle will live for less than 150 year?
(b) What is the probability that a randomly chosen turtle will live for more than 230 years?
(c) What is the probability that a randomly chose turtle will live between 160 and 210 years?
Percentiles
The term percentile is used, often incorrectly. For example, a person who is extremely tall might said to be
in the top 1% of all people, according to height. However, this does not mean that he or she is in the “first
percentile.” Instead, the person is said to be in the 99th percentile, because this measure indicates by how
many individuals the person is ahead of a particular variable. Therefore, for favourable variables such as
intelligence and happiness, it is good to be in a higher percentile. However, it is better to be in a lower
percentile for less favourable variables, such as being susceptible to a particular disease.
Example
On a college entrance exam, the scores are normally distributed. The mean is 245 and the standard
deviation is 40.
(a) If Fiona scored 300, what percentile is he in?
(b) If Jesse scored 220, what percentile is he in?