We will be generally concerned with testing hypotheses. There are many types of hypotheses that may
be of interest. Those listed below will
be introduced this semester. Note that
although our tests are usually applied to samples, we are testing hypotheses
about population parameters.
|
Z test |
H0: m
= m0
where variance (s2) is known |
|
F test |
H0: s12
= s22
|
|
Chi square test of
variance |
H0: s2
= s02
|
|
Chi square test of
independence |
H0: the
two sets of categories are independent |
|
Chi square goodness
of fit test |
H0: the
pattern of counts follow a given pattern |
|
One-sample t-test |
H0: m
= m0
|
|
Paired t-test |
H0: m
= m0
|
|
Two-sample t-test |
H0: m1
= m2
|
|
Analysis of
Variance (one-way) |
H0: m1
= m2
= m3
= m4
= … |
|
Analysis of
Variance (two-way) |
H01: m11
= m12
= m13
= … & H02: m21 = m22 = m23
= … |
|
Regression |
H0: bi = 0 |
1) Test a mean against an hypothesized
value.
a) If the variance is known, use a Z test.
b) If the variance is estimated from a sample, use a t-test.
c) The paired t-test is a special case where the two means are for
paired observations. In this case you
can take the difference between the two paired observations and test the mean
of the pairs as a single sample of differences.
2) Test a mean against another mean.
a) A two-sample t test is used to test one mean against another
mean.
b) An Analysis of Variance is used to test between means when there are
three or more means (though it can also be to test two means in which case it
gives the same result as the t-test).
c) It can also be used when means from two or more groups are to be
compared. For example, if you wish to
compare test results for class (freshman, sophomores, juniors and seniors) and
for gender (male and female). This is a
“two-way” or factorial ANOVA.
3) Testing variances.
a) A variance can be tested against an
hypothesized value with a Chi-square test.
b) Two variances can be tested for equality with the F test.
4) Chi square tests can also be used to test certain patterns of count
data.
a) They can be used to determine if numbers of something occur equally
frequently in several categories or not.
Patterns other than “equally distributed” can also be tested.
b) This test can also be used to determine if counts in a two-way table
are dependent on the table categories or not.
5) Regression is a different kind of analysis used to relate (actually
correlate) two variables. It is used to
fit the linear equation Yi = b0 + b1Xi
Examples and solutions:
1) According to published reports the
production of soybeans in northern
2) Packing crates used for tomatoes are
designed to accommodate a mean size of 3 inches with a standard deviation of no
more than 0.3 inches (variance = 0.09 inches squared). An agronomy student has developed a new
variety with a mean of 3 inches and a variance of 0.11 inches squared. He wants to determine if the variance for his
variety exceeds the required standard.
3) Mendelian genetics dictate that the
phenotypic expression of a gene with incomplete dominance should follow a 9:6:1
ratio when heterozygous individuals are crossed. The results of a genetics experiment on
butterfly color patterns show a 127:86:17 ratio. The investigator wants to determine if this
pattern departs significantly from the expected ratio.
4) A forest products student is developing
a new binder for laminated wood products.
He wants to compare the mean strength of laminated wood prepared with
his new formulation against the commercial standard. He prepares 10 pieces of laminated wood with
each binder and measures their strength.
The mean and variance of the commercial standard are known values.
5) A sociology student studying college
student spending habits. She has
interviewed several hundred students and wants to compare the mean credit card
balance for Freshmen, Sophomores, Juniors and Seniors
to determine if there are statistically significant differences.