In a relatively short amount of time, we have gone over 5 widely used statistical hypothesis tests (and the related confidence intervals). While all different in technicalities, all 5 follow the same basic format. Further hypothesis tests will also follow the "5 steps" of a hypothesis test, but the test statistics are often calculated differently, and they are not always built around the normal distribution.
Table of Parameters, Estimates, and Standard Errors
Parameter Statistic Standard Error Notes
µ 𝑥̅ 𝑠 / √𝑛 --
µ
1
- µ
2 𝑥̅
1
− 𝑥̅
2
√ 𝑠
2
1 𝑛
1
+ 𝑠
2
2 𝑛
2
When s
1
= s
2
, can use as standard error: 𝑠 𝑝
√1 𝑛
1
⁄
2
, where 𝑠 𝑝
= √
(𝑛
1
−1)𝑠
2
1
+(𝑛
2
−1)𝑠
2
2 𝑛
1
+𝑛
2
−2
µ d 𝑑̅ 𝑠 𝑑
/ √𝑛 Analysis on differences between matched pairs. p 𝑝̂ √𝑝(1 − 𝑝)/𝑛
For CI, use 𝑝̂ for p in the standard error.
For hypothesis test, use p o
for p.
When p
1
= p
2
, can use as standard error: p
1
- p
2 𝑝̂
1
− 𝑝̂
2
√ 𝑝
1
(1 − 𝑝
1 𝑛
1
)
+ 𝑝
2
(1 − 𝑝
2 𝑛
2
)
√ 𝑝(1−𝑝) 𝑛
1
+ 𝑝(1−𝑝)
, where 𝑝 = 𝑛
2 𝑥
1
+𝑥
2 𝑛
1
+𝑛
2
* NOTE: I'm fine with you using the standard errors listed in the 'standard error' column, but you book will follow the rules about when to 'pool' the standard errors (for the µ
1
- µ
2
and p
1
- p
2
cases) should you be attempting practice problems.
Step 1: Form the two hypotheses
This is the most vague step, since it depends entirely on what the researcher or employer 'wants to show'. The hypotheses are always stated in terms of the parameter (since that's what we're interested in, we already KNOW the value of the statistic). Generally speaking, whatever we are trying to prove goes into H a
, and the reverse of that goes into H o
. Also, equality should only appear in H o
. so there are three general form that the hypotheses can take:
Two-tailed Test
H o
: parameter = value
H a
: parameter ≠ value
Lower-tail Test
H o
: parameter ≥ value
H a
: parameter < value
Upper-tail Test
H o
: parameter ≤ value
H a
: parameter > value
You would fill in the proper notation for 'parameter' (µ , p , etc), and the 'value' is whatever value you want to compare to. For the 1-sample problems, this is often varies, for the 2-sample problems (difference of means, mean difference, difference of proportions -- rows 2, 3, and 5 of the table above) the value is often 0 because we are just interested in seeing if one group has a different (or larger/smaller) parameter than the other, not that one group exceeds the other by a certain amount, even though that is a perfectly valid test.
Also, you should always specify α around this time. This is given, not something you need to calculate.

Step 2: Calculate the test statistic
For all 5 of these tests, the test statistic is calculated as: Test Stat =
Estimate - Hypothesized Value
Standard Error
When dealing with proportions, the test statistic is a z and will use the standard normal distribution.
When dealing with means, the test statistic is a t, and will depend on a value called the degrees of freedom.
Test
1-sample t-test
2 independent samples t-test
(assuming equal standard deviations)
2 independent samples t-test
(NOT assuming equal standard deviations)
Degrees of Freedom (df) n - 1 n
1
+ n
2
- 2
Approximation - you won't have to calculate.*
2 paired samples t-test n - 1 ; where n is the number of pairs
* NOTE: If you are trying practice problems, the book will use the approximation (page 406) for the df.
Remember: the test statistic is like a Z-score, just a computation we do so that we can calculate probabilities.
Step 3: Calculate the p-value and/or critical value
The p-value is found by plotting the test statistic on the appropriate distribution (in these 5 tests, this will be either the t or z distribution), and then following the direction of the alternative hypothesis to find a probability.
Critical values are calculated by finding the number so that the p-value (the grey area in the graphs above) totals up to α. So for the two-tailed test, you take α/2 and find the appropriate percentile from the Z or t-distribution, for the one-tailed tests, you find the critical value based on all of α.
Step 4: Make a decision and form a conclusion
There are two ways to make a decision about whether or reject or not reject the null hypothesis (H o
).
If the p-value ≤ α , then reject H o
. Alternatively, we can say if the test statistic is 'more extreme' than the critical value, then reject H o
. By 'more extreme' we mean further out in the tails than the critical value:
Two-tailed Test
| test stat | > | Critical Value|
Lower-tail Test
Test Stat < Critical Value
Upper-tail Test
Test Stat > Critical Value
Once you make a decision about whether or not to reject Ho, write your conclusion in words. For example if you reject Ho, something to the effect of: The data provides evidence at the α level of significance that [interpret Ha in words: could be "the true mean differs from value, etc ], or if you don't reject Ho, you say the data does NOT provide evidence at the α level of significance that [interpret Ha].