Download Chapter 17: Tests of Significance

Chapter 17: Tests of Significance: The Basics
Two most common types of statistical inference tool.
1. Confidence Intervals
C.I. for Estimating a Population Mean (with σ Known)
x − E <µ< x + E
σ
where E = zα/2 · √
n
2. Tests of Significance
(Margin of error)
(Hypothesis Test)
Test of significance: A formal procedure for comparing observed data with a claim
(also called a hypothesis) whose truth we want to assess.
The step of TESTS OF SIGNIFICANCE:
STATE: What is the practical claim (question) that requires a statistical test?
PLAN: Set-up.
1. Identify the parameter.
2. State null and alternative hypotheses.
3. Choose the type of test that fits your situation.
SOLVE: Carry out the test in three phases:
1. Check the conditions for the test you plan to use.
2. Calculate the test statistic.
3. Find the P-value.
CONCLUDE: Return to the practical question to describe your results in this setting.
Note Verify three conditions.
1) The data collected “SRS” way.
2) The population is normally distributed.
3) σ known.
Procedure of a hypothesis test
Ex.
A sample of 54 bears has a mean weight of 182.9 lb. Assuming that σ is known to be 121.8 lb,
use a 0.05 significance level to test the claim that the population mean of all such bear weights
is greater than 150 lb.
1. Working with the stated claim: Read English and identify the specific claim or hypothesis.
2. Express the specific claim or hypothesis in symbolic form.
⇒ The null hypothesis: H0
A statement that the value of a population parameter is equal to some claimed value.
(Always of the form H0 : µ = a specific value)
⇒ The alternative hypothesis: Ha
We are trying to find evidence for. A contradiction of H0
Ex.
Identifying H0 and Ha . Be sure to use the correct symbol (µ) for the indicated parameter.
1. Claim: The mean annual income of employees who took a statistics course is greater
than $60,000.
2. Claim: The mean weight of plastic discarded by a household in one week is less
than 1 kg.
3. Claim: The mean pulse rete (in beats per minute) of adult females is 76.
4. Claim: The mean weight of beauty pageant winners is 108 pounds.
3. Make a figure to show the conditions for the test you plan to use.
4. What is the significance level?
”use a 0.05 significance level to test the claim”
α = 0.01, 0.05, or
0.10
Significance Level (α):
The probability of a Type I error.
Type I Error: The mistake of rejecting the null hypothesis when it is actually true.
Type II Error: The mistake of failing to reject the null hypothesis when it is actually false.
5. Calculate Test Statistics.
Test Statistics: A value used in making a decision about the null hypothesis.
: A test statistic calculated from the sample data measures how far the data
diverge from what we would expect if the null hypothesis H0 were true.
Large values of the statistic show that the data are not consistent with H0 .
Test Statistic Formulas
For mean:
(σ known)
z=
x − µ0
σ
√
n
6. P -value method:
P -value : The probability that measures the strength of the evidence against a null hypothesis.
Convert the test statistics to p-value and compare with the significance level α.
If P -value ≤ α ⇒ statistically significant at level α.
⇒ reject H0 .
If P -value > α ⇒ Not significant at Level α.
⇒ Fail to reject H0 .
The smaller the P-value, the stronger the evidence against H0 provided by the data.
Ex. Finding P-value
1. With Ha : µ < $1250, the test statistic is z = −1.23.
2. With Ha : µ > 15 ml/h, the test statistic is z = 2.88.
3. With Ha : µ ̸= 352 lb, the test statistic is z = 2.50.
7. Significant (Reject H0 ) or not significant (do not reject),
and conclusion with your English words.
There is sufficient evidence to support
the claim that ........
There is not sufficient evidence to support
the claim that ........
8. Critical value method (Traditional method):
Compere the test statistics and the critical region.
If the test statistic falls within the critical region, reject H0 .
If the test statistic does not fall within the critical region, fail to reject H0 .
9. Convert the significance level to Critical Value.
Critical Value: Any value that separates the critical
region from the noncritical region.
10. Make a figure to show Critical Region.
Critical Region: The range of values of the test value that indicates that there is
a significant difference and that the null hypothesis should be rejected.
Right-Tailed Test
H0 : µ = 150
Ha : µ > 150
Left-Tailed
H0 : µ = 150
Ha : µ < 150
Two-Tailed Test
H0 : µ = 150
Ha : µ ̸= 150
Ex.
Assume that the normal distribution applies and find the critical z value.
1. Two-tailed test; α = 0.10.
2. Left-tailed test; α = 0.10.
3. α = 0.01; Ha is µ > 0.5.
4. α = 0.005; Ha is µ ̸= 45 mm
Ex. 1
A researcher claims that the average salary of assistant technician is less than $42, 000.
A sample of 30 assistant technician has mean salary of $40, 740, and the standard deviation
of the population is $5230. (α = 0.05)
Ex. 2
A student of the author measured the sitting heights of 36 male classmate friends, and she
obtained a mean of 92.8 cm. The population of males has sitting heights with a mean of
91.4 cm and standard deviation of 3.6 cm (based on anthropometric survey data from Gordon).
Use a 0.05 significant level to test the claim that males at her college have a mean sitting height
differ from 91.4 cm.