Reject H 0 Download

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Chapter 9
Hypothesis Tests
Hypothesis Tests
The logic behind a confidence interval is that if we build
an interval around a sample value there is a high
likelihood that the population value will be captured in
the interval.
The logic behind a hypothesis test is that if we build an
interval around a hypothesized value and the sample
value falls into that interval, the result is consistent
with hypothesized value. If the sample value falls
outside the interval it is inconsistent with the
hypothesized value.
Hypotheses
Null hypothesis (H0) – The assumed value of the
population parameter when conducting the test
Alternative hypothesis (Ha) – The complement of the
null hypothesis
We don’t necessarily believe the null hypothesis is true,
it is a benchmark we are going to test against.
Forms of the Hypotheses
One-tailed test to the left
H0: m > m0
Ha: m < m0
One-tailed test to the right
H0: m < m0
Ha: m > m0
Two-tailed test
H0: m = m0
Ha: m ≠ m0
Hypothesis Tests
State of the world
Conclusion Accept H0
Reject H0
H0 is true
H0 is false
Correct
conclusion
Type II
error
Type I error Correct
conclusion
Hypothesis Tests
We can never know if the test provided the correct
answer, we can only control the probability of making a
certain type of error.
Level of significance – the probability of making a Type I
error when the null hypothesis is true as an equality.
Mechanics of a Hypothesis Test
1.
2.
3.
4.
Define the null and alternative hypotheses
Define the rule for rejecting the null hypothesis
Calculate value from sample
Reject or accept null hypothesis, state the
implication
Critical Value
A value defining the boarder(s) of the rejection
region.
Test Statistic
A value calculated from sample data that is used
to determine whether to accept or reject the null
hypothesis.
Hypothesis Test with s Known
1. H0: m > m0
Ha: m < m0
2. Reject H0 if z < -za
3.
xm
z
s
n
4. Interpret result
Hypothesis Test with s Known,
Example
A firm has started a wellness plan which
provides support for employees to lose weight
and stop smoking. In the past employees used
10 sick days per year and the standard deviation
was 2 days. The firm wants to test whether the
number of sick days has fallen with the
significance level set at 5%. A sample of 100
employees was collected, the average number of
sick days for the sample was 9.
Hypothesis Test with s Known
1. H0: m > 10
Ha: m < 10
2. Reject H0 if z < -1.645
3.
xm
9  10
1
z


 5
s
2
.2
100
n
4. Reject H0. The test implies that the number
of sick days has fallen.
Hypothesis Test with s Unknown
If s is unknown, then we use s to estimate s
and we would need to use the t distribution in
conducting the test. Again, the number of
degrees of freedom would equal n-1.
Hypothesis Test with s Unknown,
Example
A researcher wants to know if the amount workers
are putting into tax sheltered retirement plans (like
IRA and 401k accounts) has fallen. Last year the
average monthly contribution into a tax-sheltered
plan was $150. A sample of 101 workers is drawn.
The average monthly contribution for the workers
in the sample was $135, and the sample standard
deviation was $50. Conduct the test with a 5%
significance level.
Hypothesis Test with s Unknown
1. H0: m > 150
Ha: m < 150
2. Reject H0 if t < -1.660
3.
x  m0 135  150  15
t


 3.01
s
50
4.975
n
101
4. Reject H0. The evidence suggests that the
amount of money contributed to taxsheltered programs has fallen.
Hypothesis Test with s Unknown,
Example
Last year Medicare recipients spent an average
of 4.5 days in the hospital. A researcher wants to
test to see if the average has changed. He wants
to conduct a test at the 5% significance level. A
sample of 25 Medicare recipients is drawn and
the sample mean is 5 days and the standard
deviation is 2.
Hypothesis Test with s Unknown
1. H0: m = 4.5
Ha: m ≠ 4.5
2. Reject H0 if t < -2.064 or t > 2.064
x  m0 5  4.5 .5
3.
t

  1.25
s
2
.4
25
n
4. Accept H0. The evidence is not strong enough
to reject H0, we would assume the average
number of days in the hospital has not
changed.
Alternatives Ways of Conducting a
Hypothesis Test
• t scores
• p values
p Values
A p value is the probability of drawing a value
whose distance from the hypothesized value is
greater than or equal to the sample value.
The null hypothesis is rejected if p < a. In a onetailed test p is equal to the area in a single tail. In
the case of a two-tailed test p equals the area in
both tails that is as far away from the mean as the
sample value.
Comparisons of Rejection Rules
Assume a one-tailed test in which a = .05
and z equals 2.
Hypothesis Test using p-Value, s
Unknown
Last year Medicare recipients spent an average
of 4.5 days in the hospital. A researcher wants to
test to see if the average has changed. He wants
to conduct a test at the 5% significance level. A
sample of 25 Medicare recipients is drawn and
the sample mean is 5 days and the standard
deviation is 2.
Hypothesis Test using p-Value,
s Unknown
1. H0: m = 4.5
Ha: m ≠ 4.5
2. Reject H0 if p < .05
x  m0 5  4.5 .5
3.
t

  1.25
s
2
.4
25
n
p  P(t  1.25) or P(t  1.25)  .22
4. Accept H0. The evidence is not strong enough to
reject H0, we would assume the average number
of days in the hospital has not changed.
Hypothesis Test with s Unknown,
Example
A researcher wants to know if the amount workers
are putting into tax sheltered retirement plans (like
IRA and 401k accounts) has fallen. Last year the
average monthly contribution into a tax-sheltered
plan was $150. A sample of 101 workers is drawn.
The average monthly contribution for the workers
in the sample was $135, and the sample standard
deviation was $50. Conduct the test with a 5%
significance level.
Hypothesis Test with s Unknown
1. H0: m > 150
Ha: m < 150
2. Reject H0 if p < .05
3.
x  m0 135  150  15
t


 3.01
s
50
4.975
n
101
p  P(t  3.01)  .00165
4. Reject H0. The evidence suggests that the
amount of money contributed to taxsheltered programs has fallen.
Summary of Rejection Rules
z and t scores: The rejection rule is defined in
terms of standard deviations
p values: The rejection rule is defined in terms of
probabilities
Summary of Hypothesis Tests
About a Population Mean
Lower Tail Test Upper Tail Test Two-Tailed Test
t score
rejection rule
H0: m > m0
Ha: m < m0
Reject H0 if
t<-ta
H0: m < m0
Ha: m > m0
Reject H0 if
t>ta
H0: m = m0
Ha: m ≠ m0
Reject H0 if
t<-ta/2 or t>ta/2
p-value
rejection rule
Reject H0 if
p<a*
Reject H0 if
p<a*
Reject H0 if
p < a **
Hypotheses
* Where p is calculated using one tail
** Where p is calculated using both tails