Download Conducting a One Sample Test of Hypothesis

Examples of One Sample
Test of Hypothesis
Testing Manufacturer’s Claims
• X-15 steel-belted radial truck tire
• Specs:
• Lifetime of product (tread)
• 60,000 miles +/- 5000 miles
• Finley Trucking:
• Purchased 48 tires for fleet.
• Substantial wear after 58,000
miles!
Is Finley’s experience with the tires
consistent with the claims of the
manufacturer?
What we know
• The manufacturer claims the mean life is 60,000 miles
• The standard deviation is 5,000 miles
• Finley Trucking bought 48 tires
• The mean life of the tires Finley purchased was 58,000 miles
• Step 1: State the Hypothesis
• There is no significant difference from the manufacturer’s claims
and the experience of Finley Trucking. That is, the mean is at least as good as
the manufacturer claims, and perhaps even better.
Ho: μ ≥ 60000
• The alternative is that the mean is less than 60,000 miles.
Ha: μ < 60000
• Step 2: Select a level of significance
Probability of rejecting Ho, when it is actually true.
Also known as a Type I Error
α = 5% = 0.05
(we want this to be as low as practical)
• Step 3: State the Test Statistic
• Things to consider:
• Is the population standard deviation known?
• Yes: use the Standard Normal Probability Distribution
• No: use the Student’s t Probability Distribution
Population Standard Deviation Known
• Step 4: Formulate the Decision
Rule
• A statement of the specific conditions
that must be met for the null
hypothesis, Ho, to be rejected.
• One-Tailed Test
• α = 0.05
• σ is known.
• Rejection region for a one-tailed test
is always in the tail “pointed at” by
the Ha.
Ha: u < 60000, so…
Reject Ho if Z < -1.645
• Step 5. Compute the value of the test statistic
• Step 6: Make a decision
• REJECT Ho
• The calculated value of the test statistic was less than the critical value of Z
• Step 7: Interpret the Results
There is sufficient statistical evidence at the 5% significance level
to reject the null hypothesis, in favor of the alternative
hypothesis that the mean tread life of the X-15 steel-belted
radial truck tire is less than 60,000 miles.
Ace Carts: Is there any difference is assembly
times?
On average it takes 42.3 minutes
to assemble the J2+2Sport
New method:
6 carts assembled
mean = 44.5 minutes
std.dev. = 2.7 minutes
Step 1. State the hypothesis
Ho: μ = 42.3 minutes
Ha: μ ≠ 42.3 minutes
two-tailed test!
Step 2. Level of Significance
α = 0.05
Step 3. Test Statistic
μ is not known
Step 4. Rejection region
• Two-Tailed Test, σ unknown
• Significance level
• α = 0.05
• Degrees of Freedom
• n=6
• df = n - 1 = 5
• Critical value:
• 2.571
Reject Ho if t < -2.571, or if t > 2.571
Step 5: Calculate the test statistic
On average it takes 42.3 minutes
to assemble the J2+2Sport
New method:
6 carts assembled
mean = 44.5 minutes
std.dev. = 2.7 minutes
Step 6. Decision
Fail to reject Ho
Why?
The calculated value of the test
statistic, 1.996, does not fall
within either rejection region.
Ace Carts: Is there any difference is assembly
times?
There is insufficient statistical
evidence at the 5% significance
level to reject the null hypothesis
that the new assembly time is
the same as the old assembly
time of 42.3 minutes.