One Sample Z-test
• Convert raw scores to z-scores to test
hypotheses about sample
• Using z-scores allows us to match z with a
probability
• Calculate:
z = (statistic – parameter)/standard error of statistic
= (sample mean – population mean)/standard error of mean
where standard error of mean = s/sqrt(n)
One Sample Z-test
• If the calculated Z falls outside of the “likely”
zone of our distribution, we reject the null
hypothesis.
• The area in the distribution that falls outside of
the “likely” zone is called the “region of
rejection” and is formally called a --- or the
probability of making a Type 1 error
• Typically a is set at .05
a
a
Types of alternative hypotheses
• Non-directional – “there is a difference”
– Also called “two-tailed”
• Directional – “there is a difference that is
directional”
– Also called “one-tailed”
• Students in CPS have a different average GRE
score than OSU graduate students in general
• Students in CPS have a higher average GRE
score than OSU graduate students in general
Types of alternative hypotheses
• Students in CPS have a different average GRE
score than OSU graduate students in general
• H0: m=average OSU GRE
• H1:m  average OSU GRE
• Students in CPS have a higher average GRE
score than OSU graduate students in general
• H0: m=average OSU GRE
• H1:m > average OSU GRE
Two tailed hypothesis
• If we take a sample of 30 students and
find their average IQ to be 106.58, we can
test whether the population from which our
sample came had an average IQ of 100
H0: m=100 (average IQ)
H1:m ≠ 100
• Convert our sample mean to Z statistic
(106.58-100)/2.74 = 2.4
Z=-2.4
Z= 2.4
One tailed hypothesis
• If we take a sample of 30 students and find their average
IQ to be 106.58, we can test whether the population from
which our sample came had an average IQ greater than
100
H0: m=100 (average IQ)
H1:m > 100
• Convert our sample mean to Z statistic
(106.58-100)/2.74 = 2.4
Z=2.4
One tailed hypothesis
• If we take a sample of 30 students and find their average
IQ to be 93.42, we can test whether the population from
which our sample came had an average IQ less than
100
H0: m=100 (average IQ)
H1:m < 100
• Convert our sample mean to Z statistic
(93.42-100)/2.74 = - 2.4
Z=-2.4
Steps of Hypothesis Testing
1.
2.
3.
4.
State Hypothesis
Set Criterion for Rejection H0
Compute Test Statistic and Probability
Make Decision
One-Sample T-test
• If we do not know the s of the population –
we can’t calculate the SE of the mean
• We can estimate it with the SD of the
sample: SEmean=s/sqrt(n)
• Instead of calculating z statistic – we
calculate the t-statistic
t = (statistic – parameter)/standard error of statistic
= (sample mean – population mean)/standard error of mean
where standard error of mean = s/sqrt(n)
One-Sample T-test
• After calculating t-statistic, we find
associated p-value
• SPSS does this for us
• T is about the same as Z in large samples.
• For this class we will always use T
Making a decision
• We compare the calculated p-value to our
preset a level
• If p< a, reject null hypothesis
• If p> a, fail to reject null hypothesis
1-tailed t-test
1-tailed t-test
1-tailed t-test
1-sample t-test
• As sample size increases, the results from
a t-test approximate a z-test
• SPSS only does one sample t-tests