Transcript Hypothesis Testing For a Single Population Mean

Hypothesis Testing
For a Single Population Mean
Example: Grade inflation?
Population of
5 million college
students
Sample of
100 college students
Is the average
GPA 2.7?
How likely is it that
100 students would
have an average
GPA as large as 2.9
if the population
average was 2.7?
The p-value illustrated
How likely is it that 100 students would have an average
GPA as large as 2.9 if the population average was 2.7?
Determining the p-value
H0: μ = average population GPA = 2.7
HA: μ = average population GPA > 2.7
If 100 students have average GPA of 2.9 with standard
deviation of 0.6, the P-value is:
P( X  2.9)  P[Z  (2.9  2.7) /(0.6 / 100)]
 P[Z  3.33]  0.0004
Making the decision
• The p-value is “small.” It is unlikely that
we would get a sample as large as 2.9 if the
average GPA of the population was 2.7.
• Reject H0. There is sufficient evidence to
conclude that the average GPA is greater
than 2.7.
Terminology
• H0: μ = 2.7 versus HA: μ > 2.7 is called a
“right-tailed” or a “one-sided” hypothesis
test, since the p-value is in the right tail.
• Z = 3.33 is called the “test statistic”.
• If we think our p-value is small if it is less
than 0.05, then the probability that we make
a Type I error is 0.05. This is called the
“significance level” of the test. We say,
α=0.05, where α is “alpha”.
Example: Body Temperature?
Population of
many, many adults
Sample of
80 adults
Is average adult
body temperature
98.6 degrees? Or
is it lower?
Average body
temperature of 80
sampled adults is
98.4 degrees.
The p-value illustrated
How likely is it that 80 adults would have an average body
temp as small as 98.4 if the pop’n average was 98.6?
Determining the p-value
H0: μ = average pop’n body temp = 98.6
HA: μ = average pop’n body temp < 98.6
If 80 adults have average body temp of 98.4 with
standard deviation of 0.6, the P-value is:
P( X  98.4)  P[Z  (98.4  98.6) /(0.6 / 80 )]
 P[Z  2.98]  0.001
Making the decision
• The p-value is “small.” It is unlikely that
we would get a sample as small as 98.4 if
the average body temp of the population
was 98.6.
• Reject H0. There is sufficient evidence to
conclude that the average body temp is
smaller than 98.6.
Terminology
• H0: μ = 98.6 versus HA: μ < 98.6 is called a
“left-tailed” or a “one-sided” hypothesis
test, since the p-value is in the left tail.
• Z = -2.98 is the “test statistic”.
• If we think our p-value is small if it is less
than 0.02, then the probability that we make
a Type I error is 0.02. That is, significance
level α = 0.02.
Example: $ on Alcohol?
Population of
Penn State students
Is average amount
spent weekly $20?
Sample of
64 students
Average amount
spent is $17 with
standard deviation
of $16.
The p-value illustrated
How likely is it that 64 students would spend an average as
small as $17, or as large as $23, if the pop’n avg was $20?
Determining the p-value
H0: μ = average $ spent = $20
HA: μ = average $ spent  $20
If 64 students spend an average of $17 with standard
deviation of $16, the P-value is:
P( X 17)  P[Z  (17  20) /(16 / 64 )]
 P[Z  1.5]  0.067
and
P( X  23)  0.067
So P-value = 0.067  2 = 0.134
Making the decision
• The p-value is not “small.” It is likely that
we would get a sample as small as $17, or
as large as $23, if the average amount spent
on alcohol was $20.
• Do not reject H0. There is not enough
evidence to conclude that the average
amount spent differs from $20.
Terminology
• H0: μ = $20 versus HA: μ  $20 is called a
“two-tailed” or a “two-sided” hypothesis
test, since the p-value is in both tails.
• Z = -1.5 is the “test statistic”.
• Since we failed to reject the null hypothesis,
we may have made a Type II error.
Very Important Point
• Your p-value will not be correct unless
the assumptions are correct!!!!
• You must have a large sample to use the
methods presented here.