Download Hypothesis testing

Hypothesis testing
March 20, 2000
Two-tailed tests
• A two-tailed
test means
that an equal
part of the
standard error
is in each “tail”
of the
distribution.
Confidence interval
• Using z scores, we
can calculate the
actual values
represented by the
boundary of the
area in the tails.
• For example, if we
want a 95%
confidence limit, we
find the values for
which 2.5% of
scores (half of 5%)
fall below and 2.5%
fall above.
confidence interval
• Homework review
Hypothesis testing with sample data
Do two variables have a nonrandom
relationship?
• Four basic steps
– State the null and research hypothesis
– Select a significance level (alpha ())
– Select and compute a test statistic
– Make a decision by comparing to critical
value of the test statistic.
Example - t test
• Meier/Brudney, 11.12
• Change in toll booths from human collection
to machine collection.
• Human attendants (population)
•  = 1,253 cars per hour
• New system; sample size of 100 hours
•  = 1,261 cars per hour
st dev = 59
• State a hypothesis, null hypothesis, and
evaluate them. What is your conclusion?
• You are comparing the sample mean to the
population mean.
Example - t test
• State the research and null hypothesis.
• H1: The average number of cars passing
through automated booths is higher than
the average number of cars passing
through booths with human attendants.
• H0:There is no difference in the average
number of cars passing through booths
with automated collections and those with
human attendant collection.
• Note: this is a directional hypothesis. It
states that one is higher or lower than
the other. Use a one-tailed test.
Example - t test
• Compute the statistic
X
t
s n
1261  1253
8
8
t

  305
.
.
59 100 59 10 59
Example - t test
• Find the critical t
• Because this problem has a directional
hypothesis, we will use a one-tailed
test.
– The degrees of freedom are equal to the
sample size minus 1.
– Using the t table, find the critical t for  =
.05 and df = 99.
• The critical t = 1.66
Example - t test
• So, the calculated t = 3.05
• the critical t (t.05) = 1.66
• Because the calculated t score is greater than
the critical t, we reject the null hypothesis.
• There is a statistically significant difference in
the mean number of cars passing through the
automated tool booths and the mean number
of cars passing through booths with human
attendants.
T-test
• Practice problems
One sample z test
• We use the t-test when we want to
compare the means of a sample to a
population or the means of two samples
if we know the mean and standard
deviation for the samples, but only the
mean of the population.
• If, however, we also know the standard
deviation of the population, we use a
one-sample z test.
One sample z test
• Don’t confuse with z scores.
– One-sample z test is an inferential statistic.
– Z score is a descriptive statistic.
Example - one sample z test
• Over the course of many years, a
Midwestern college has a mean law
school acceptance rate of 50% per year.
The standard deviation = 3.5
• The pre-law advisor developed a
program intended to improve the rate
of acceptance in 1999.
• In 1999, 130 graduates applied to law
school; 73 were accepted (56%).
• Did law school acceptances improve
following the program?
Example - one sample z test
• State the research and null hypotheses.
• H1: Law school admission rates were
higher for those participating in a pre-law
advising program.
• H0:There is no difference in law school
admission rates between those who
participate in an advising program and
those who do not.
• Note: This is a one-tailed hypothesis. One
mean is higher than the other.
Example - one sample z test
• Select an .
• Remember, in social sciences the general
standard is  = .05 (remember, this
equates to what percent of the time you
are likely to be wrong).
• Choose a test statistic - since we are
comparing the mean of a group
(sample) to a known population mean
and standard deviation, the one sample
z test is an appropriate statistic.
Example - one sample z test
• Compute the statistic. Note: the denominator
is the formula for a standard error.
sample mean - pop mean
pop st dev number of cases in sample
z
X 


n
Example - one sample z test
• Calculate z
56  50
6
6
z


 19.35
. 114
.
0.31
35
. 130 35
Critical value of z
(two-tailed test for non-directional Ho)
Probability
(level of significance)
Critical value of z
0.05
1.96
0.01
2.58
0.001
3.29
Critical value of z
(one-tailed test for directional Ho)
Probability
(level of significance)
Critical value of z
0.05
1.65
0.01
2.33
0.001
3.09
Example - one sample z test
• Calculated z = 19.35
• Critical z = 1.65
– (from z table,  = .05) (z.05 = 1.65)
• Since the calculated z is > z.05 , we
reject the null hypothesis. The law
school admission rate is statistically
significantly higher following the
advising program.
one sample z test
• Practice problems