Download 8.5 Test of Hypothesis about a Population Mean

Inferences Based on a Single Sample:
Tests of Hypothesis
8.1 The Elements of a Test of
Hypothesis
Statistical hypotheses – statements about population
parameters
Examples
mean weight of adult males is greater than 160
proportion of students with a 4.0 GPA is less than .01
In statistics, we test one hypothesis against another
The hypothesis that we want to prove is called the
alternative hypothesis, H a
Another hypothesis is formed that contradicts H a . This
hypothesis is called the null hypothesis, H 0
After taking the sample, we must either: Reject H 0 and
believe H a , or Fail to reject H 0 because there was not
sufficient evidence to reject it (meaning there is not
sufficient evidence to conclude H a )
Types of errors
H 0 is true
H 0 is false
Fail to reject H 0
Correct
Reject H 0
Type I error
Type II error
Correct
The probability with which we are willing to risk a type I
error is called the level of significance of a test
P(reject H 0 | H 0 is true )  
The probability of making a type II error is
P(fail to reject H 0 | H 0 is false)  
The quantity1   is known as the power of a test. It
represents the probability of rejecting H 0 when in fact
it is false.

Decreasing
increases  which decreases power
Increasing sample size increases power
Test Statistic – the statistic we compute to make the
decision. Sampling distribution of test statistic given
that H 0 is true must be known or well approximated
Critical region (Rejection region) – the values of the test
statistic such that we reject H 0 and conclude that H a is
true
Critical value – the endpoint of the critical region
8.2 Formulating Hypotheses and Setting
Up the Rejection Region
Steps of a hypothesis test
1) State H a and H 0
2) Specify ( , n, and) critical region
3) (Take sample and) compute test statistic
4) Make decision
5) Interpret the results
* Information in parentheses will be given in problems
8.3 Test of Hypothesis about a
Population Mean: Normal (z) Statistic
Example
It is claimed that the mean score for elementary
education majors on a test of mathematical
competency is less than 35.
Test the claim at   .05
Assume n  165, x  32.91, and   9.5
Conclude with 95% confidence that the mean score for
elementary education majors on the test of
mathematical competency is less than 35.
8.4 Observed Significance Levels:
p-values
The choice of  is subjective
The p-value of a hypothesis test is the smallest value of 
such that H 0 would have been rejected
If p - value   , Reject H 0
If p - value   , Fail to reject H 0
Example
Go back to the mathematical competency of elementary
education majors.
x
32.91  35

 2.83
We calculated Z 

9.5
n
165
8.5 Test of Hypothesis about a
Population Mean: Student’s t-Statistic
Example
A consumer protection agency wants to prove that
packages of Post Grape Nuts have an average weight
that is not 24 oz.
  .05
n  100
x  23.94
s  .13
Conclude with 95% confidence that packages of Post
Grape Nuts have a population mean weight not equal
to 24 oz.
Give a 95% confidence interval using the Post Grape
Nuts information.
We are 95% confident that the population mean weight
of Post Grape Nuts packages is between 23.914 and
23.966 oz.
Example
It is desired to prove that the population mean weight of
metal components produced by a process is greater
than 4.5 oz.
  .05
n  10
x  4.59
s  .504
There is not sufficient evidence to prove that the
population mean weight of the metal components is
greater than 4.5 oz.
Example
A marketing consultant wants to prove that the
population mean household income in Lexington is
less than $50,000.
  .05
n  100
x  40,571
s  8,316
Conclude with 95% confidence that the mean household
income in Lexington is less than $50,000.
Example
A farmer has experienced an average weight gain in his
pigs of 200 lbs over a fixed time. He is experimenting
with a new feeding technique for his pigs and wants to
know if the average weight gain will change in either
direction.
x  197.2
  .05
n  24
s  9.8
There is not sufficient evidence to prove the average
weight gain of the pigs is different from 200 lbs using
the new feeding technique.
8.6 Large-Sample Test of Hypothesis
about a Population Proportion
Example
The current treatment for a type of cancer produces
remission 20% of the time. An investigator wishes to
prove that a new method is better. Suppose 26 of 100
patients go into remission using the new method.
  .05
There is not sufficient evidence to conclude the new
method is better.
Example
Do less than 50% of people prefer Murray’s Vanilla
Wafer’s when compared to other brands? Suppose 42
of the 250 chose Murray’s.
  .05
Conclude with 95% confidence that less than 50% of
people prefer Murray’s Vanilla Wafer’s when compared
to other brands.