Download What`s a Statistical Hypothesis?

What’s a Statistical Hypothesis?
A statistical hypothesis is
a statement about the
numerical value of a
population parameter.
I believe the mean GPA of
this class is 3.5!
© 1984-1994 T/Maker Co.
Hypothesis Testing for Population Mean
Population
☺
☺
☺ ☺
☺
☺
I believe the
population mean
age is 50
(hypothesis).
☺
Random
sample
Mean ☺
☺X = 20
Reject
hypothesis!
Not close.
Null Hypothesis
• The null hypothesis, denoted H0, represents the
hypothesis that will be “retained” unless the data provide
convincing evidence that it is false. This usually represents
the “status quo” or some claim about the population
parameter that the researcher wants to test.
• You may think of null hypothesis as the “favored”
hypothesis; we reject it in favor of the alternative
hypothesis Ha if and only if the evidence provided by the
sample data are strong against H0 and in favor of Ha .
• “retain H0” is commonly referred to as “do not reject”.
• Stated in one of the following forms
H0: µ = (some
value) Book uses this version…
(
H0: µ ≤ (some
value)
(
H0: µ ≥ (some
value)
(
Alternative Hypothesis
1. Opposite of null hypothesis
2. The hypothesis that will be accepted only if
the data provide convincing evidence of its
truth
3. Designated Ha
4. Stated in one of the following forms
Ha: µ ≠ (some
value)
(
Ha: µ < (some
value)
(
Ha: µ > (some
value)
(
Identifying Hypotheses
Example 1: If the hypothesis of a researcher is that
the population mean is not 3, set-up the hypotheses
to be tested.
Steps:
• State the question statistically
µ≠3
• State the opposite statistically
µ=3
• State the null hypothesis statistically
H0: µ = 3
• State the alternative hypothesis statistically
Ha: µ ≠ 3
Identifying Hypotheses
Example 2: If the hypothesis of a researcher is that
the population mean is greater than 3, set-up the
hypotheses to be tested.
Steps:
• State the question statistically
µ>3
• State the opposite statistically
µ≤3
• State the null hypothesis statistically
H0: µ ≤ 3 or µ = 3
• State the alternative hypothesis statistically
Ha: µ > 3
Identifying Hypotheses
Example 3: Is the population average amount of TV
viewing 12 hours?
• State the question statistically: µ = 12
• State the opposite statistically: µ ≠ 12
• Select the alternative hypothesis: Ha: µ ≠ 12
• State the null hypothesis: H0: µ = 12
Identifying Hypotheses
Example 4: Is the population average amount of TV
viewing different from 12 hours?
• State the question statistically: µ ≠ 12
• State the opposite statistically: µ = 12
• Select the alternative hypothesis: Ha: µ ≠ 12
• State the null hypothesis: H0: µ = 12
Identifying Hypotheses
Example 5: Is the average cost per hat less than or
equal to $20?
• State the question statistically: µ ≤ 20
• State the opposite statistically: µ > 20
• Select the alternative hypothesis: Ha: µ > 20
• State the null hypothesis: H0: µ ≤ 20 or
H0: µ = 20
Identifying Hypotheses
Example 6: Is the average amount spent in the
bookstore greater than $25?
• State the question statistically: µ > 25
• State the opposite statistically: µ ≤ 25
• Select the alternative hypothesis: Ha: µ > 25
• State the null hypothesis: H0: µ ≤ 25
H0: µ = 25
Test Statistic
The test statistic is a sample statistic,
computed from information provided in the
sample, that the researcher uses to decide
between the null and alternative hypotheses.
Hypothesized value for
population mean
Student’s t-statistic
If population standard deviation σ is unknown use
sample standard deviation s.
Assumptions for using Student’s t
statistic
• The data values should be independent.
• The data arise from a random sample.
• The population has a relative frequency
distribution that is approximately normal.
Type I Error
• A Type I error occurs if the researcher rejects
the null hypothesis in favor of the alternative
hypothesis when, in fact, H0 is true.
• The probability of committing a Type I error is
denoted by α.
• It is also called “level of significance”
Type II Error
A Type II error occurs if the researcher retains
the null hypothesis when, in fact, H0 is false.
The probability of committing a Type II error is
denoted by β.
Conclusions and Consequences for
a Test of Hypothesis
True State of Nature
Conclusion
H0 True
Ha True
Do not reject H0
Correct decision Type II error
(Assume H0 True)
(probability β)
Type I error
Reject H0
(Assume Ha True) (probability α)
Correct decision
• How will we decide if we reject the
null hypothesis?
Example
• Lets assume we would like to test
H0: µ =2400 against Ha: µ < 2400
• So we will reject the null hypothesis if our
sample mean takes a value which is far below
2400.
• Lets assume sample mean=2000
Basic Idea
Sampling Distribution
It is unlikely
that we
would get a
sample
mean of this
value ...
If P(sample mean <2000) is
very small, then we can
conclude that the sample mean
2000 is not the result of just
chance, then we reject
Η0 :µ = 2400.
... if in fact this were
the population mean
2000
Area= P(sample mean <2000)
µ = 2400
H0
Sample Means
• Instead of working with smple mean, we use the test statistic
to decide.
• Test statistic can be thought as standardized version of the
sample statistic.
• The sample mean is the sample statistic for population mean.
• Instead of trying to determine P(sample mean <2000) we will
try to determine if P(t n-1<test statistic) is small.
Basic Idea
Sampling Distribution for t-statistics
If P(T<t) is very small,
It is unlikely
then we reject Η0 :µ =
that we
would get a
2400.
sample
mean of this
value ...
... if in fact this were
the population mean
test statistic- t
Area= P( T <t)=p-value
µ=0
H0
Sample Means
p-Value
•
•
•
•
•
Probability of obtaining a test statistic more
extreme (≤ or ≥) than actual sample value,
given H0 is true
Can be thought of as a measure of the
“credibility” of the null hypothesisH0 .
α is the nominal level of significance. This value
is assumed by an analyst.
p-value is also probability for making type-I
error.
But, p-value is called “observed level of
significance”.
• If p-value ≥ α, do not reject H0
• If p-value < α, reject H0
• The p-value shows our confidence to reject null
hypothesis.
• If this value is smaller than α, then the probability
that we will reject null hypothesis when it is true
is even smaller than the maximum tolerated error
probability α.
• So we can conclude that null hypothesis is wrong
and can be rejected in favor of alternative
hypothesis.
• The smaller the p-value is, the more confident we
are with our decision to reject H0 .
Steps for Calculating the p-Value
Step 1:
Determine the value of the test statistic t
corresponding to the result of the sampling
experiment.
Steps for Calculating the p-Value
Step 2:
• Lower-tailed test ( Ha:µ< µ0)
p-value=P(t n-1 < test statistic)
• Upper-tailed test ( Ha: µ> µ0 )
p-value=P(t n-1 > test statistic)
• Two-tailed test ( Ha: µ≠ µ0 )
p-value=2P(t n-1 > |test statistic|)
Reporting Test Results as
p-Values: How to Decide Whether to
Reject H0
1. Choose the maximum value of α that you are
willing to tolerate.
2. If the observed significance level (p-value) of the
test is less than the chosen value of α, reject the
null hypothesis. Otherwise, do not reject the
null hypothesis.
3. Typical values for α are 0.01, 0.05, 0.10.
Example 1
Does an average box of cereal
contain 368 grams of cereal? A
random sample of 25 boxes
showed x = 372.5. The
company has specified σ to be
15 grams. Find the p-value.
How does it compare to α =
.05?
368 gm.
Example 2
Does an average box of cereal
contain more than 368 grams
of cereal? A random sample
of 25 boxes showed x = 372.5.
The company has specified σ
to be 15 grams. Find the pvalue. How does it compare
to α = .05?
368 gm.
Exercise 1
You’re an analyst for Ford. You
want to find out if the average
miles per gallon of Escorts is less
than 32 mpg. You take a sample
of 60 Escorts & compute a sample
mean of 30.7 mpg and sample
standard deviaiton of 3.8 mpg.
What is the p-value? How does it
compare to α = .01?
Exercise 2
Is the average capacity of
batteries less than 140
ampere-hours? A random
sample of 20 batteries had a
mean of 138.47 and a standard
deviation of 2.66. Assume a
normal distribution. Test at the
.05 level of significance.
Exercise 3
Does an average box of
cereal contain 368 grams of
cereal? A random sample
of 25 boxes had a mean of
372.5 and a standard
deviation of 12 grams. Test
at the .05 level of
significance.
368 gm.
Exercise 4
You work for the FTC. A manufacturer
of detergent claims that the mean
weight of detergent is 3.25 lb. You
take a random sample of 16
containers. You calculate the sample
average to be 3.238 lb. with a
standard deviation of .117 lb. At the
.01 level of significance, is the
manufacturer correct?
3.25 lb.