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Section 1.3: The Normal Distribution
Normal curves are N( μ , σ )
- Bell-shaped, unimodal, symmetric
- Mean, μ, always center of the graph
- Standard deviation, σ, controls the spread of the graph, where curve changes direction
- Probabilities are just the area under the curve (integral) between the points of interest
In the last set of Chapter 1 notes, we discussed bell-shaped distributions, standardization, and the 68-95-99.7% rule.
Standard Normal Distribution
What if you need different probabilities for X ∼ N(μ, σ)? Do we have to use Calculus? No. We have a great shortcut—the Normal table, Table A, in the front cover of your book.
You must convert X ∼ N(μ, σ) to Z ∼ N(0, 1), where Z has the standard Normal distribution. Convert using the formula:
x Z
Z -scores are what you need in order to use Table A in the front cover of your book.
Z -scores also let you compare 2 values from different Normal distributions to see their probabilities on the same scale.
P ( Z < z -score) is what you will find on the Normal table. What if you want to know something else?
- P(Z > z-score) = 1 – P(Z≤ z-score)
- P(Z = z-score) = 0. Therefore
P Z ( ≤ z − score ) = P Z ( < z − score ) and P Z ( ≥ z − score ) = P Z ( > z − score )
- P(a < Z < b) = P(Z < b) – P(Z < a).
Z -scores tell you how far (measured in standard deviations) the original observations fall from the mean.
To find a probability if you have X ∼ N( μ , σ ) and a sample score to work with:
- Convert X to Z.
x Z
- Rearrange (if necessary) the inequality so that it uses < or ≤. Remember that P(Z > z-score) = 1 – P(Z ≤ z-score).
- Look up the probability for your z-score on Table A.
- If z -score is between 2 table values, either pick the closer one or average the two closest values.
Example: Practice using the Normal table with some generic problems where we already know Z (instead of starting from X ). Find the probabilities.
a) P ( Z < 1.24 )
b) P ( Z > 1.24 )
c) P ( Z < -1.24 )
d) P ( Z > -1.24)
e) P (Z = 1.24 )
f) P ( -1.24 < Z < 1.24)
g) P ( Z < -4.5 )
h) P ( Z < 12 )
Example: Now practice using the Normal table with some generic problems where we start with X and then have to find Z before going to the table. For all the problems below, use μ = 2 and σ = 1.2.
a) P ( X < 1 )
b) P ( X > 4 )
c) P ( 0 < X < 3.4)
“Backwards” Normal Problems If you are given the probability and know X ∼ N(μ, σ), but you don’t know the sample’s score (backwards from the previous problems):
- Treat it as P(Z < z-score) = the probability. Work backward from the probability in Table A to a corresponding z-score.
- Adjust to < if necessary by doing the “1 –“ trick.
- If you have a 2-sided probability, use P(-z 0 < Z < z 0 ) = 2 P(Z < z-score) – 1.
- Convert the z-score to x by converting with x = μ + zσ.
Example: Practice using the normal table with some generic backwards-normal problems where we know the probability and are just looking for z 0 (instead of going all the way to x 0 ). Find the probabilities.
a) P ( Z < z 0 ) = 0.
b) P ( Z < z 0 ) = 0.
c) P ( Z < z 0 ) = 0.
d) P (Z < z 0 ) = 0.
e) P ( Z > z 0 ) = 0.
f) P ( Z > z 0 ) = 0.
g) P ( -z 0 < Z < z 0 ) = 0.
Example: Now practice using the normal table with some generic backwards-normal problems where we know the probabilities and are now going all the way to x 0 by way of z 0. For all the problems below, use μ = 2 and σ = 1.2.
a) P ( X < x 0 ) = 0.
b) P (X > x 0 ) = 0.
c) Find the 2 values between which 20% of the data lies.
Example: In the checking account example where the balances are ~ N (1325, 25),
a) What is the account balance, x 0 , such that the percentage of balances less than it is 23%?
b) What is the account balance, x 0 , such that the probability of a balance being more than it is 0.15?
c) What is the account balance, x 0 , such that the probability of a balance being more than it is 0.5? (Hint: you should be able to do this one without math.)
d) Between what 2 central values do 40% of the balances fall?
b. What is the probability that a randomly selected student’s total SAT score will be less than 1000?
c. What was the cut-off for the top 10% of total SAT scores?
d. What is the mean and standard deviation of the difference between a student’s math and verbal scores (math - verbal)?
e. What is the probability that a randomly selected student will score at least 20 points higher on the verbal part of the exam than on the math part?
f. Do you think that our assumption that the verbal and math scores are independent is reasonable? Why or why not?
g.
