How do we know which test/formula to use for a hypothesis question?
After we have learned Chapters 8, 9, 10, 12 and 13, which are mostly about hypothesis, since there are so many
different tests/formulas, we might have a question “ How do we know which test/formula should be applied to a
hypothesis question?”. I think a helpful method is to clarify the differences among the tests/formulas and also
make a tree chart.
“The methods presented in Chapters 8,9,10 and 12 are called parametric methods because they are based on
sampling from a population with specific parameters (A parameter is a numerical measurement describing some
characteristic of a population.), such as the mean, standard deviation or proportion. Those parametric methods
usually must conform to some fairly strict conditions, such as a requirement that the sample data come from a
normally distributed population.” (“ELEMENTARY STATSTICS, 13-1, OVERVIEW) On the other hand, nonparametric
methods which do not have such strict requirements are introduced in Chapter 13. Therefore, if we see sentences
such as “Do not assume normality.” Or “Assume the population is not necessarily normally distributed.” in a
question, we know that the question is from Chapter 13.
Here is the tree chart of a summary of tests/formulas for Chapters 8 to 13 except Chapter 7:
One Group: Chapter 8 σ known: z test, 𝑧 =
(Is σ known?)
𝜎
√𝑛
σ unknown but s known: t test, 𝑡 =
Two Groups: Chapter 9
Is the data
𝑥−µ
Two Independent
𝑠
√𝑛
σ known: z test,
𝑧=
(Are the data independent
𝑥−µ
(𝑥1 −𝑥2 )−(µ1 −µ2 )
𝜎 ² 𝜎 ²
√ 1+ 2
𝑛1
parametric or
Parametric samples or matched Pairs?) (Is σ known?)
σ unknown but s known: t test,
𝑡=
nonparametric? (How many
(𝑥1 −𝑥2 )−(µ1 −µ2 )
𝑠 ² 𝑠 ²
√ 1+2
𝑛1
Matched Pairs: 𝑡 𝑡𝑒𝑠𝑡, 𝑡 =
groups of
samples are Three and More: Chapter 12
in the
question?)
𝑛2
𝑛2
𝑑− µ𝑑
𝑠𝑑
√𝑛
One Factor: One-Way ANOVA
(How many factors are in the question?) Two Factors: Two-Way ANOVA
Usually use Excel: Data  Data Analysis  One-Way ANOVA or Two-Way ANOVA)
Correlation and Regression, Chapter 10: r: linear correlation coefficient, r²: coefficient of
determination, linear regression equation: 𝑦̂ = b0 + b1x
Two Groups
Repeated:
(Are the data
Sign Test or Wilcoxon Signed-Ranks Test
repeated or
Note: Reject the null hypothesis if the test statistic value is less than
or equal to the critical value found in Table A-8 .
independent?)
Two Independent Samples:
Wilcoxon Rank-Sum Test
Nonparametric, Chapter 13
Three or More: Kruskal-Wallis Test
(How many groups of
samples are in the question?)
Spearman Rank
No Ties: 𝑟𝑠 = 1 −
Correlation
Ties: 𝑟𝑠 =
(Does either variable have
ties among its ranks?)
6 ∑ 𝑑²
𝑛(𝑛2 −1)
𝑛 ∑ 𝑥𝑦−(∑ 𝑥)(∑ 𝑦)
√𝑛(∑ 𝑥²)−(∑ 𝑥)²√𝑛(∑ 𝑦²)−(∑ 𝑦)²
Can you figure out which formula/test to apply to the following question by using the formula/test tree chart?
Example:
A random sample of nine students was selected to test for the effectiveness of a special course designed to
improve memory. The following table gives the results of a memory test given to these students before and after
the course.
Before
43 57 48 65 81 49 38 69 58
After
49 56 55 77 89 57 36 64 69
a. Test at the 1% significance level whether this course makes any statistically significant improvement in the
memory of all students. Assume the population has a normal distribution.
b. Repeat this question assuming that the population is not necessarily normally distributed.
Question a.
Step one: Parametric or nonparametric?
The sentence “Assume the population has a normal distribution.” tells us that it is parametric.
Step two: How many groups of samples are there?
There are two groups of samples there (This type of question is from chapter 9).
Step three: Independent or matched pairs?
The words “before” and “after” show that this question is about matched pairs.
Therefore, we use t test, 𝑡 =
𝑑− µ𝑑
𝑠𝑑
√𝑛
for parametric matched pairs, a formula which is in Chapter 9.
Question b.
Step one: Parametric or nonparametric?
The words “assuming that the population is not necessarily normally distributed” indicate that it is nonparametric. Right away, you
know that this question type belongs to Chapter 13.
Step two: How many groups of samples?
There are two groups of samples.
Step three: Independent or matched pairs?
The words “before” and “after” show that this question is about matched pairs.
If you find Sign Test or Wilcoxon Signed-Ranks Test, CONGRATULATIONS! This is the right test for this question.