Download 10.2.1 Steps of a Hypothesis Test

Testing Claims about a Population Mean
Page 1
Testing a Claim Regarding a Population Mean
Step 0: Verify Assumptions
The hypothesis test of a population mean has two assumptions.
1. The sample is obtained using simple random sampling
2. The sample has no outliers and the population from which the sample is drawn is
normally distributed or the sample size, n, is large (n ≥ 30).
Step 1: State the Hypothesis
A claim is made regarding the population mean. This claim is used to determine the null and
alternative hypotheses. The hypotheses can be structured in one of the following ways:
Two-Tailed
H0: µ = µ0
H1: µ ≠ µ0
Left-Tailed
H0: µ ≥ µ0
H1: µ < µ0
Right-Tailed
H0: µ ≤ µ0
H1: µ > µ0
Step 2: Select a Level of Significance
The selection of the level of significance α is done based on the seriousness of making a Type I
error. (The typical value of α is 0.05.)
Step 3: Calculate the Test Statistic
The test statistic represents the number of standard deviations the sample mean is from the
claimed population mean, µ0. When σ is known, then a z-value may be found. When σ is
unknown, then a t-value must be found
z-Test
x − µ0
z0 =
t-Test
x − µ0
t0 =
s
n
σ
n
Step 4 (The Classical Approach): Find the Critical Value
The level of significance is used to determine the critical value, represented by the t-values in the
figures below. The critical region includes the values of the shaded region. The shaded region is
α.
Two-Tailed
−t α / 2
Robert A. Powers
tα / 2
Left-Tailed
−tα
Right-Tailed
tα
University of Northern Colorado
Testing Claims about a Population Mean
Page 2
Step 4 (The Modern Approach): Find the p-Value
Based on the critical value t0, determine the probability that a sample mean is further from the
mean than is hypothesized. This is represented by the shaded region in the figures below.
Two-Tailed
P(T < -|t0| or T > |t0|)
− | t0 |
| t0 |
Left-Tailed
P(T < -t0)
− t0
Right-Tailed
P(T > t0)
t0
Step 5: Make a Decision
Reject the null hypothesis if the test statistics lies in the critical region or the probability
associated with the test statistic is less than the level of significance.
Do not reject the null hypothesis if the test statistic does not lie in the critical region or the
probability associated with the test statistic is greater than or equal to the level of significance.
Step 6: State the Conclusion
State the conclusion of the hypothesis test based on the decision made and with respect to the
original claim.
Reject H0
Do Not
Reject H0
Original Claim is H0
There is sufficient evidence (at the α
level) to reject the claim that … .
There is not sufficient evidence (at the
α level) to reject the claim that … .
Robert A. Powers
Original Claim is H1
There is sufficient evidence (at the α
level) to support the claim that … .
There is not sufficient evidence (at the
α level) to support the claim that … .
University of Northern Colorado