μ=dfμ=df and σ=2⋅dfσ=2⋅df
- Use goodness-of-fit to test whether a data set fits a particular
probability distribution.
- The degrees of freedom are number of cells or categories - 1number of cells or categories - 1.
- The test statistic is
Σ
k
(
O
−
E
)
2
E
Σ
k
(
O
−
E
)
2
E
, where
OO = observed values (data),
EE = expected values (from theory), and
kk = the number of different data cells
or categories.
- The test is right-tailed.
- Use the test of independence to test whether two factors are
independent or not.
- The degrees of freedom are equal to
(number of columns - 1)(number of rows - 1)(number of columns - 1)(number of rows - 1).
- The test statistic is
Σ
(
i
⋅
j
)
(
O
-
E
)
2
E
Σ
(
i
⋅
j
)
(
O
-
E
)
2
E
where
OO = observed values,
EE = expected values,
ii = the number of rows in the
table, and
jj = the number of columns in
the table.
- The test is right-tailed.
- If the null hypothesis is true, the expected number
E
=
(row total)(column total)
total surveyed
E=
(row total)(column total)
total surveyed
.
- Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution as each other.
- The degrees of freedom are equal to
number of columns - 1number of columns - 1.
- The test statistic is
Σ
(
i
⋅
j
)
(
O
-
E
)
2
E
Σ
(
i
⋅
j
)
(
O
-
E
)
2
E
where
OO = observed values,
EE = expected values,
ii = the number of rows in the
table, and
jj = the number of columns in
the table.
- The test is right-tailed.
- If the null hypothesis is true, the expected number
E
=
(row total)(column total)
total surveyed
E=
(row total)(column total)
total surveyed
.
The expected value for each cell needs to be at least 5 in order to
use the Goodness-of-Fit, Independence and Homogeneity tests.
- Use the test to determine variation.
- The degrees of freedom are the number of samples - 1.
- The test statistic is
(
n
-
1
)
⋅
s
2
σ
2
(
n
-
1
)
⋅
s
2
σ
2
, where nn = the total number of data, s2s2 = sample variance, and σ2σ2 = population variance.
- The test may be left, right, or two-tailed.
(What does "Endorsed by" mean?)
"Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"