http://www.ck12.org Chapter 8. Hypothesis Testing
Standard Deviation:
sd=√
∑(d^2 )−(∑d)2
n
n− 1=
√
254 −(^32 )
2
9
8≈ 4. 19
Standard Error of the Difference:
sd ̄=
sd
√
n=
4. 19
√
9
= 1. 40
Test Statistic(t-Test)
t=d ̄−δ
sd ̄=
3. 6 − 0
1. 40
≈ 2. 57
With 8 degrees of freedom (number of observations - 1) and a significance level of.05, we find our critical values to
be 2.306 standard scores above and below the mean. Since our test statistic of 2.57 exceeds this critical value, we
canreject the null hypothesisthat the two samples are equal and conclude that the lesson had an effect on student
achievement.
Lesson Summary
- In addition to testing single samples associated with a mean, we can also perform hypothesis tests with two
samples. We can test two independent samples (which are samples that do not affect one another) or dependent
samples which assume that the samples are related to each other. - When testing a hypothesis about two independent samples, we follow a similar process as when testing one
random sample. However, when computing the test statistic, we need to calculate the estimated standard error of the
difference between sample means which is found by using the formula:
sX ̄ 1 −X ̄ 2 =
√
s^2(
1
n 1+
1
n 2)
, wheres^2 =nss 1 +^1 +n 2 ss−^22- We carry out the test of two independent samples in a similar way as the testing of one random sample. However,
we use the following formula to calculate the test statistic:
t=(
X ̄ 1 −X ̄ 2 )−(μ 1 −μ 2 )
sX ̄ 1 −X ̄ 2 , wheresX ̄ 1 −X ̄ 2 =
√
s^2 (1
n 1+
1
n 2)
- We can also test the proportions associated with two independent samples. In order to calculate the test statistic
associated with two independent samples, we use the formula:
z=
(p 1 −p 2 )−(P 1 −P 2 )
sp 1 −p 2- We can also test the likelihood that two dependent samples are related. To calculate the test statistic for two
dependent samples, we use the formula:
t=
d ̄−δ
sd ̄