STA 2023 Chapter 6 Calculator Instructions:
Probabilities
Normal Distribution
The procedure for finding probabilities of values from any normally distributed
population are the same as those for finding probabilities from standard normal
distribution.
Example:
Find the probability of getting a score of 32 given the distribution is normally distributed
with a mean of 45 and a standard deviation of 12.
Using the normalpdf( function and typing in the score
32 with a mean of 45 and standard deviation of 12
we get the probability of .018.
Find the probability of getting a score less than 32 given the distribution is normally
distributed with a mean of 45 and a standard deviation of 12.
Using the normalcdf( function and typing in -1E99,
then the score 32 with a mean of 45 and standard
deviation of 12 we get the probability .139.
Example: Soda Cans
Suppose that the contents (in ounces) of a soda can labeled “12 ounces” follows a normal
distribution, with mean 12 and standard deviation 0.015.
(a) What is the probability that the contents of a can selected at random lie
between 11.97 and 11.99 ounces?
normalcdf(11.97, 11.99, 12, 0.015) = 0.2297.
(b) What is the probability that the contents of a can selected at random lie
between 12.02 and 12.07 ounces?
normalcdf(12.02, 12.07, 12, 0.015) = 0.0912.
Example: Calculator Warranties
A calculator has a normally distributed life-span (in months) with mean µ = 54 and
standard deviation σ = 8. The calculator’s warranty will replace a defective calculator up
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Normal Distribution Probabilities (continued)
to 36 months old. What percentage of the calculators produced will be replaced under this
warranty? We want to find the probability of the calculator lasting under 36 months:
normalcdf(−1E99, 36, 54, 8) = 0.0122 = 1.22%.
Inverse Normal Distribution Probabilities
There are many times in statistics where we are given a probability, and have to find a
relevant z-score or raw score. Such problems are known as inverse normal distribution
problems. Again, such computations can be performed using tables of normal
probabilities, but the work is tedious, error prone, and full of rounding errors.
Fortunately the calculator has a function, invNorm( , that performs the calculation. The
invNorm function is located on the same DISTR page that normalcdf( is.
Example: Calculator Warranties Revisited
We looked earlier at a warranty for a calculator with a normally distributed life-time in
months with mean 54 and standard deviation 8. How long should the warranty offer
replacement for defective calculators if we want to replace at most 1% of the calculators?
To find the score that is associated with the lowest 1% of the area under the normal
distribution we use invNorm( .
Press the 2nd key and then the VARS key to get to the
DISTR page.
Press the number 3 key.
Type in .01, 54, 8).
Press the ENTER key.
We should offer to replace defective calculators up to
35 months old.
Example: SAT Scores
Given that the 2002 SAT scores were normally distributed with mean 1020 and standard
deviation 153, what is the 90th percentile score?
invNorm(0.90, 1020, 153) = 1216.
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