Using the Standard Normal Curve


1.   P(0 < z < k) = Prob. at k from table
2.   P(k1 < z < k2) = P(0 < z < k2) - P(0 < z < k1)
3.   P(z > k) = 0.5 - P(0 < z < k)
4.   P(z < k) = 0.5 + P(0 < z < k)
5.   P(-k < z < 0) = P(0 < z < k)   See Rule 1.
6.   P(-k1 < z < k2) = P(-k1 < z < 0) + P(0 < z < k2)
= P(0 < z < k1) + P(0 < z < k2)
7.   P(-k < z < k) = 2P(0 < z < k)
8.   P(z < -k) = P(z > k)   See Rule 3.
9.   P(-k1 < z < -k2) = P(k2 < z < k1)   See Rule 2.
10.   P(z > -k) = P(-k < z < 0) + P(z > 0)
= P(0 < z < k) + 0.5


Examples to Accompany Rules


1. Find P(0 < z < 1.37).

rule1.jpe (23072 bytes)

P(0 < z < 1.37) = 0.4147


2. Find P(1.28 < z < 2.46)

rule2.jpe (22241 bytes)

P(1.28 < z < 2.46) = P(0 < z < 2.46) - P(0 < z < 1.28)
= 0.4931 - 0.3997 = 0.0934


3. Find P(z > 2.50).

rule3.jpe (20889 bytes)

P(z > 2.50) = P(z > 0) - P(0 < z < 2.50)
= 0.5 - 0.4938 = 0.0062


4. Find P(z < 0.83).

rule4.jpe (24038 bytes)

P(z < 0.83) = P(z < 0) + P(0 < z < 0.83)
= 0.5 + 0.2967 = 0.7967


5. Find P(-1.78 < z < 0).

rule5.jpe (23474 bytes)

P(-1.78 < z < 0) = P(0 < z < 1.78)
= 0.4625


6. Find P(-2.21 < z < 1.02).

rule6.jpe (24177 bytes)

P(-2.21 < z < 1.02) = P(-2.21 < z < 0) + P(0 < z < 1.02)
= P(0 < z < 2.21) + P(0 < z < 1.02)
= 0.4864 + 0.3461 = 0.8325


7. Find P(-1.35 < z < 1.35).

rule7.jpe (23326 bytes)

P(-1.35 < z < 1.35) = P(-1.35 < z < 0) + P(0 < z < 1.35)
= P(0 < z < 1.35) + P(0 < z < 1.35)
= 2P(0 < z < 1.35)
= 2(0.4115) = 0.8230


8. Find P(z < -0.93).

rule8.jpe (22105 bytes)

P(z < -0.93) = P(z > 0.93)
= P(z > 0) - P(0 < z < 0.93)
= 0.5 - 0.3238 = 0.1762


9. Find P(-1.25 < z < -0.77).

rule9.jpe (22706 bytes)

P(-1.25 < z < -0.77) = P(0.77 < z < 1.25)
= P(0 < z < 1.25) - P(0 < z < 0.77)
= 0.3944 - 0.2794 = 0.1150


10. Find P(z > -2.64).

rule10.jpe (25121 bytes)

P(z > -2.64) = P(-2.64 < z < 0) + P(z > 0)
= P(0 < z < 2.64) + 0.5
= 0.4959 + 0.5 + 0.9959


The Four Step Procedure for Tests of Hypothesis


  1. Set up the null and alternate hypotheses
  2. Calculate a test statistic
  3. Construct a Rejection Region
  4. Draw a Conclusion

Let's elaborate on these! We will consider tests of hypotheses of the mean of one population for Large Samples; i.e., n > 30.

  1. Set up the null and alternate hypotheses
    A H0: µ ³ µ0 vs. H1: µ < µ0. Key words: less than, decrease, diminish, reduce, at least.
    B H0: µ £ µ0 vs. H1: µ > µ0. Key words: greater than, increase, exceeds, at most.
    C H0: µ = µ0 vs. H1: µ µ0. Key words: equal to, the same, any/no change, any/no difference.
  2. Calcuate a test statistic
    Z* =
    x - µ0
    s
    n
  3. Construct a Rejection Region
    a. If H1 has either < or >, then you have a 1 tail test. Hence, you take 0.5 - a and find the horizontal z value that corresponds to that probability.
    b. If H1 has , then you have a 2 tail test. Now take 0.5 - a/2, and find the horizontal z value that corresponds to that probability.
    c. Write an If/then statement: If |Z*| > Z, then we reject H0.
  4. Draw a Conclusion: Reject H0 or fail to reject H0.


For Small Samples, Steps 1 and 4 are identical. The differences appear in Steps 2 and 3.

2. Calculate a test statistic
t* =
x - µ0
s
n
3. Construct a Rejection Region
a. If H1 has either < or >, then you have a 1 tail test. Hence, you find ta, n-1.
b. If H1 has , then you have a 2 tail test. Now find ta/2,n-1.
c. Write an If/then statement: If |t*| > t, then we reject H0.

This document was translated from LATEX by HEVEA.