Download §9.3--Hypothesis Tests for One Population Mean When is Known

§9.3–Hypothesis Tests for One Population Mean
When σ is Known
Tom Lewis
Fall Term 2009
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
1/9
Outline
1
An overview of the method
2
Find critical values
3
Some examples
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
2/9
An overview of the method
The key idea
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea
Throughout this section we will be formulating simple hypothesis
tests for one population mean.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea
Throughout this section we will be formulating simple hypothesis
tests for one population mean.
In these tests, we will consider only one population. In each case we
will testing whether the true mean µ of the population is equal to a
hypothesized mean µ0 . Thus the null hypothesis will always be of the
form H0 : µ = µ0 .
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea
Throughout this section we will be formulating simple hypothesis
tests for one population mean.
In these tests, we will consider only one population. In each case we
will testing whether the true mean µ of the population is equal to a
hypothesized mean µ0 . Thus the null hypothesis will always be of the
form H0 : µ = µ0 .
The alternative hypothesis will be of one of three forms: left-tailed,
two-tailed, or right-tailed.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea
Throughout this section we will be formulating simple hypothesis
tests for one population mean.
In these tests, we will consider only one population. In each case we
will testing whether the true mean µ of the population is equal to a
hypothesized mean µ0 . Thus the null hypothesis will always be of the
form H0 : µ = µ0 .
The alternative hypothesis will be of one of three forms: left-tailed,
two-tailed, or right-tailed.
We are assuming that the null hypothesis is true and we are looking
for evidence to the contrary. We will reject H0 only when the evidence
suggests that something improbable has occurred. The significance
level α sets the threshold for improbability.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea
Throughout this section we will be formulating simple hypothesis
tests for one population mean.
In these tests, we will consider only one population. In each case we
will testing whether the true mean µ of the population is equal to a
hypothesized mean µ0 . Thus the null hypothesis will always be of the
form H0 : µ = µ0 .
The alternative hypothesis will be of one of three forms: left-tailed,
two-tailed, or right-tailed.
We are assuming that the null hypothesis is true and we are looking
for evidence to the contrary. We will reject H0 only when the evidence
suggests that something improbable has occurred. The significance
level α sets the threshold for improbability.
A random sample of size n is collected and its sample mean, x, is
computed.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea
Throughout this section we will be formulating simple hypothesis
tests for one population mean.
In these tests, we will consider only one population. In each case we
will testing whether the true mean µ of the population is equal to a
hypothesized mean µ0 . Thus the null hypothesis will always be of the
form H0 : µ = µ0 .
The alternative hypothesis will be of one of three forms: left-tailed,
two-tailed, or right-tailed.
We are assuming that the null hypothesis is true and we are looking
for evidence to the contrary. We will reject H0 only when the evidence
suggests that something improbable has occurred. The significance
level α sets the threshold for improbability.
A random sample of size n is collected and its sample mean, x, is
computed.
Continued on next slide.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
3/9
An overview of the method
The key idea continued
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
4/9
An overview of the method
The key idea continued
The key idea is this: under the null hypothesis, the test statistic
√
z = (x − µ0 )/(σ/ n) is a standard normal random variable.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
4/9
An overview of the method
The key idea continued
The key idea is this: under the null hypothesis, the test statistic
√
z = (x − µ0 )/(σ/ n) is a standard normal random variable.
The logic of the test is this. Since z has a standard normal
distribution, its value should be close to 0. Values of z that are 2 and
3 units distance from 0 are therefore improbable and create suspicion.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
4/9
An overview of the method
The key idea continued
The key idea is this: under the null hypothesis, the test statistic
√
z = (x − µ0 )/(σ/ n) is a standard normal random variable.
The logic of the test is this. Since z has a standard normal
distribution, its value should be close to 0. Values of z that are 2 and
3 units distance from 0 are therefore improbable and create suspicion.
Depending upon the form of the alternative hypothesis and the
significance level of the test, a non-rejection and a rejection region are
determined.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
4/9
An overview of the method
The key idea continued
The key idea is this: under the null hypothesis, the test statistic
√
z = (x − µ0 )/(σ/ n) is a standard normal random variable.
The logic of the test is this. Since z has a standard normal
distribution, its value should be close to 0. Values of z that are 2 and
3 units distance from 0 are therefore improbable and create suspicion.
Depending upon the form of the alternative hypothesis and the
significance level of the test, a non-rejection and a rejection region are
determined.
If z falls into the non-rejection region, then we assume that there is
not enough evidence against the null hypothesis to reject it.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
4/9
An overview of the method
The key idea continued
The key idea is this: under the null hypothesis, the test statistic
√
z = (x − µ0 )/(σ/ n) is a standard normal random variable.
The logic of the test is this. Since z has a standard normal
distribution, its value should be close to 0. Values of z that are 2 and
3 units distance from 0 are therefore improbable and create suspicion.
Depending upon the form of the alternative hypothesis and the
significance level of the test, a non-rejection and a rejection region are
determined.
If z falls into the non-rejection region, then we assume that there is
not enough evidence against the null hypothesis to reject it.
If z falls into the rejection region, then we are declaring that it is
better to reject H0 than to accept that something as improbable as z
has just occurred by random sampling.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
4/9
Find critical values
Problem
A hypothesis test is being conducted to test H0 : µ = µ0 , where µ is the
true population mean. The population standard deviation σ is known. A
random sample of size n is found, and x and the test statistic
√
z = (x − µ0 )/(σ/ n) are computed. In each of the following, find the
critical value(s) and rejection and non-rejection regions.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
5/9
Find critical values
Problem
A hypothesis test is being conducted to test H0 : µ = µ0 , where µ is the
true population mean. The population standard deviation σ is known. A
random sample of size n is found, and x and the test statistic
√
z = (x − µ0 )/(σ/ n) are computed. In each of the following, find the
critical value(s) and rejection and non-rejection regions.
Ha : µ < µ0 and α = .05
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
5/9
Find critical values
Problem
A hypothesis test is being conducted to test H0 : µ = µ0 , where µ is the
true population mean. The population standard deviation σ is known. A
random sample of size n is found, and x and the test statistic
√
z = (x − µ0 )/(σ/ n) are computed. In each of the following, find the
critical value(s) and rejection and non-rejection regions.
Ha : µ < µ0 and α = .05
Ha : µ > µ0 and α = .05
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
5/9
Find critical values
Problem
A hypothesis test is being conducted to test H0 : µ = µ0 , where µ is the
true population mean. The population standard deviation σ is known. A
random sample of size n is found, and x and the test statistic
√
z = (x − µ0 )/(σ/ n) are computed. In each of the following, find the
critical value(s) and rejection and non-rejection regions.
Ha : µ < µ0 and α = .05
Ha : µ > µ0 and α = .05
Ha : µ 6= µ0 and α = .05
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
5/9
Find critical values
Problem
A hypothesis test is being conducted to test H0 : µ = µ0 , where µ is the
true population mean. The population standard deviation σ is known. A
random sample of size n is found, and x and the test statistic
√
z = (x − µ0 )/(σ/ n) are computed. In each of the following, find the
critical value(s) and rejection and non-rejection regions.
Ha : µ < µ0 and α = .05
Ha : µ > µ0 and α = .05
Ha : µ 6= µ0 and α = .05
Ha : µ < µ0 and α = .1
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
5/9
Find critical values
Problem
A hypothesis test is being conducted to test H0 : µ = µ0 , where µ is the
true population mean. The population standard deviation σ is known. A
random sample of size n is found, and x and the test statistic
√
z = (x − µ0 )/(σ/ n) are computed. In each of the following, find the
critical value(s) and rejection and non-rejection regions.
Ha : µ < µ0 and α = .05
Ha : µ > µ0 and α = .05
Ha : µ 6= µ0 and α = .05
Ha : µ < µ0 and α = .1
Ha : µ 6= µ0 and α = .1
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
5/9
Some examples
Problem
Courtesy of a University of North Carolina statistics web site.
An insurance company is reviewing its current policy rates. When
originally setting the rates they believed that the average claim amount
was $1,800. They are concerned that the true mean is actually higher than
this, because they could potentially lose a lot of money. They randomly
select 40 claims, and calculate a sample mean of $1,950. Assume that the
standard deviation of claims is $500, and set α = 05, and test to see if the
insurance company should be concerned.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
6/9
Some examples
Solution
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Solution
Let µ represent the true mean of insurance payouts. Let the null
hypothesis be H0 : µ = 1800.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Solution
Let µ represent the true mean of insurance payouts. Let the null
hypothesis be H0 : µ = 1800.
In this case, the insurance company is only concerned about the true
mean exceeding 1800, since this leads to a loss of revenue. Therefore
we formulate this as a right-tailed test with Ha : µ > 1800.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Solution
Let µ represent the true mean of insurance payouts. Let the null
hypothesis be H0 : µ = 1800.
In this case, the insurance company is only concerned about the true
mean exceeding 1800, since this leads to a loss of revenue. Therefore
we formulate this as a right-tailed test with Ha : µ > 1800.
√
Under H0 , the test statistic z = (x − 1800)/(500/ 40) has a
standard normal distribution.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Solution
Let µ represent the true mean of insurance payouts. Let the null
hypothesis be H0 : µ = 1800.
In this case, the insurance company is only concerned about the true
mean exceeding 1800, since this leads to a loss of revenue. Therefore
we formulate this as a right-tailed test with Ha : µ > 1800.
√
Under H0 , the test statistic z = (x − 1800)/(500/ 40) has a
standard normal distribution.
Since our test is right-tailed, we will reject H0 only if the test statistic,
z, is large and positive. At α = .05, the critical value is z.05 = 1.645.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Solution
Let µ represent the true mean of insurance payouts. Let the null
hypothesis be H0 : µ = 1800.
In this case, the insurance company is only concerned about the true
mean exceeding 1800, since this leads to a loss of revenue. Therefore
we formulate this as a right-tailed test with Ha : µ > 1800.
√
Under H0 , the test statistic z = (x − 1800)/(500/ 40) has a
standard normal distribution.
Since our test is right-tailed, we will reject H0 only if the test statistic,
z, is large and positive. At α = .05, the critical value is z.05 = 1.645.
Our non-rejection region is therefore (−∞, 1.645) and our rejection
region is (1.645, ∞).
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Solution
Let µ represent the true mean of insurance payouts. Let the null
hypothesis be H0 : µ = 1800.
In this case, the insurance company is only concerned about the true
mean exceeding 1800, since this leads to a loss of revenue. Therefore
we formulate this as a right-tailed test with Ha : µ > 1800.
√
Under H0 , the test statistic z = (x − 1800)/(500/ 40) has a
standard normal distribution.
Since our test is right-tailed, we will reject H0 only if the test statistic,
z, is large and positive. At α = .05, the critical value is z.05 = 1.645.
Our non-rejection region is therefore (−∞, 1.645) and our rejection
region is (1.645, ∞).
For the sample collected, z = 1.89. Since z falls into the reject
region, we will choose to reject H0 at this significance level.
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
7/9
Some examples
Problem
A family practice clinic claims that the amount of time a doctor spends
with a patient is normally distributed with a mean of 14.5 minutes and a
standard deviation of 1.5 minutes. A patient watchdog group, hearing
complaints about patients being rushed through the clinic, randomly
samples 25 patients and finds that their average visit was 14.14 minutes.
At a significance level of α = .05, can they charge the clinic with
misrepresenting the lengths of their office visits?
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
8/9
Some examples
Problem
Tests of older baseballs showed that when dropped 24 ft onto a concrete
surface, they bounced an average of 235.8 cm. In a test of 40 new
randomly selected baseballs, the bounce heights had a mean of 235.4 cm.
Assume that the standard deviation for bounce heights is 4.5 cm. Use a
α = .05 significance level to test the claim that the mean bounce heights
of the new baseballs is different from 235.8 cm. Are the new baseballs
different?
Tom Lewis ()
§9.3–Hypothesis Tests for One Population Mean When σ is Known
Fall Term 2009
9/9