(One group of samples with 𝝈 known)
Step one: State the null and alternative hypotheses
𝐻𝑜 : 𝑀 = 𝜇
𝐻1 : 𝑀 ≠ 𝜇
Step two: Find the critical Z value by the significance level 𝜶 and the number of tails.
Step three: Collect data and compute the test statistic Z.
1.
𝜎𝑀 =
2.
𝑍=
𝜎
√𝑛
𝑀−𝜇
𝜎2
𝑜𝑟 𝜎𝑀 = √ 𝑛
𝜎𝑀
Step four: Make a decision.
Method one:
If the absolute value of the test statistic value > the absolute value of the critical value, we reject the null hypothesis and the P-value < the
significance level𝛼.
If the absolute value of the test statistic value < the absolute value of the critical value, we fail to reject the null hypothesis and the P-value >
the significance level 𝛼.
Method two:
If the test statistic falls in the critical region, we reject the null hypothesis and the P-value < the significance level α.
If the test statistic does not fall in the critical region, we fail to reject the null hypothesis and the P-value > the significance level α.
(One group of samples with 𝝈 unknown but s known)
Step one: State the null and alternative hypotheses
𝐻𝑜 : 𝑀 = 𝜇
𝐻1 : 𝑀 ≠ 𝜇
Step two: Find the critical t value by the significance level 𝜶, the degree of freedom df=n-1 and the number of tails.
Step three: Collect data and compute the test statistic t.
1. 𝑆𝑆 = ∑ 𝑋 2 − (∑ 𝑋)2 /𝑛
𝑠𝑠
𝑆𝑆
2. 𝑆 = √𝑛−1
3.
𝑆𝑀 =
4. 𝑡 =
𝑆
√𝑛
𝑜𝑟 𝑆 2 = 𝑛−1
𝑆2
or 𝑆𝑀 = √ 𝑛
𝑀−𝜇
𝑆𝑀
Step four: Make a decision
(One group of samples did the same test twice: before and after; repeated; or
matched pairs)
Step one: State the null and alternative hypotheses
𝐻𝑜 : 𝜇𝐷 = 0
𝐻1 : 𝜇𝐷 ≠ 0
Step two: Find the critical t value by the significance level 𝜶, the degree of freedom df=n-1 and the number of tails.
Step three: Collect data and compute the test statistic t.
1. 𝑆𝑆 = ∑ 𝐷2 − (∑ 𝐷)2 /𝑛
𝑠𝑠
2. 𝑆 = √𝑛−1
3. 𝑆𝑀𝐷 =
4. 𝑡 =
𝑆
√𝑛
𝑀𝐷 −𝜇𝐷
or
𝑆𝑆
𝑆 2 = 𝑛−1
𝑆2
or 𝑆𝑀𝐷 = √ 𝑛
𝑆𝑀𝐷
Step four: Make a decision
(Two independent groups of samples)
Step one: State the hypothesis.
Ho: 𝜇1 = 𝜇2 (The sign can be “<,≤, > 𝑜𝑟 ≥” )
H1: 𝜇1 ≠ 𝜇2 (Same as above.)
Step two: Find the critical t value.
The critical t value is decided by the significance level 𝛼; the degree of freedom𝑑𝑓 = (𝑛1 − 1) + (𝑛2 − 1) and the number
of tails.
Step three: Collect the data and compute the test statistic.
1. Sum of Squares: 𝑆𝑆1 = ∑𝑋1 2 − (∑𝑋1 )2 /𝑛1
𝑆𝑆2 = ∑𝑋22 − (∑𝑋2 )2 /𝑛2
𝑆𝑆 +𝑆𝑆
2. Pooled Variance: 𝑆𝑝 2 = 𝑑𝑓1 +𝑑𝑓2 = (𝑛
1
2
𝑆𝑆1 +𝑆𝑆2
1 −1)+(𝑛2 −1)
𝑆𝑝 2
𝑆𝑝 2
1
𝑛2
3. Estimated standard error: 𝑆(𝑀1 −𝑀2 ) = √ 𝑛 +
4. t statistic: 𝑡 =
(𝑀1 −𝑀2 )−(𝜇1 −𝜇2 )
𝑆(𝑀1 −𝑀2 )
Step Four: Make a conclusion.
Note:
If we wouldn’t reject Ho in two-tailed test, we cannot reject Ho in one-tailed test.
In SPSS print out, sig. # is the p-value for two-tailed test because SPSS only has two-tailed test. If it is one-tail test, to find p-value, the sig.
number should time 2.
Critic al value
Chapter 8:
One sample Z test
𝝈 𝐤𝐧𝐨𝐰𝐧
Z critical value is decided
by the significance level α
and the one or two tail(s)
Hypothesis
(example)
𝐻𝑜 : 𝑀 = 𝜇
𝐻1 : 𝑀 ≠ 𝜇
Or
𝐻𝑜 : 𝜇 = 50
𝐻1 : 𝜇 ≠ 50
Chapter 9:
One sample t-test
𝝈 unknown but s known
Chapter 11:
Dependent samples t-test
t critical value is decided
by the significance level α,
the one or two tail(s), and
the df=n-1
t critical value is decided
by the significance level α,
the one or two tail(s), and
the df = n-1 (n is the
number of differences.)
Chapter 10:
t critical value is decided
Two independent samples by the significance level α,
the one or two tail(s), and
the 𝒅𝒇 = 𝒅𝒇𝟏+ 𝒅𝒇𝟐 =
(𝒏𝟏 − 𝟏) + (𝒏𝟐 − 𝟏).
Estimated
standard error
𝜎
𝜎𝑀 =
√𝑛
Or
𝑍=
𝑀−𝜇
𝜎𝑀
𝑡=
𝑀−𝜇
𝑆𝑀
𝑡=
𝑀𝐷 − 𝜇𝐷
𝑆𝑀𝐷
𝜎2
𝜎𝑀 = √
𝑛
𝑆
𝐻𝑜 : 𝑀 = 𝜇
𝐻1 : 𝑀 ≠ 𝜇
𝑆𝑀 =
Or
Or
𝐻𝑜 : 𝜇 = 50
𝐻1 : 𝜇 ≠ 50
𝑆2
𝑆𝑀 = √
𝑛
𝐻𝑜 : 𝜇𝐷 = 0
𝐻1 : 𝜇𝐷 ≠ 0
𝑆𝑀𝐷 =
Or
√𝑛
𝑆
√𝑛
𝑆2
𝑆𝑀𝐷 = √
𝑛
𝐻𝒐 : 𝜇1 = 𝜇2
𝐻1 : 𝜇1 ≠ 𝜇2
Test statistic formula
𝑆(𝑀1 −𝑀2 )
𝑆𝑝 2 𝑆𝑝 2
=√
+
𝑛1
𝑛2
(𝜇𝐷 = 0)
𝑡=
(𝑀1 − 𝑀2 ) − (𝜇1 − 𝜇2 )
𝑆(𝑀1 −𝑀2)
(𝜇1 − 𝜇2 = 0)
When to use estimation: When a hypothesis test leads to a rejection of the null hypothesis (Ho)
1. One sample t-statistic: 𝜇 = 𝑀 ± 𝑡(𝑆𝑀 )
Conclusion (example): We are 90% confident that the population mean (𝜇) lies within the interval from … to …
2. Independent samples t-statistic: 𝜇1 − 𝜇2 = (𝑀1 − 𝑀2 ) ± 𝑡(𝑆(𝑀1 −𝑀2 ) )
Conclusion (example): We are 90% confident that the difference of the population means lies within the interval from … to …
3. Dependent samples t-statistic: 𝜇𝐷 = 𝑀𝐷 ± 𝑡(𝑆𝑀𝐷 )