Chapter 8 Section 4.ppt

Section 8.4-1
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

Section 8.4-2
Chapter 8
Hypothesis Testing
8-1 Review and Preview
8-2 Basics of Hypothesis Testing
8-3 Testing a Claim about a Proportion
8-4 Testing a Claim About a Mean
8-5 Testing a Claim About a Standard Deviation or
Variance

Section 8.4-3
Key Concept
This section presents methods for testing a claim about a
population mean.
Part 1 deals with the very realistic and commonly used
case in which the population standard deviation σ is not
known.
Part 2 discusses the procedure when σ is known, which is
very rare.

Section 8.4-4
Part 1
When σ is not known, we use a “t test” that incorporates
the Student t distribution.

Section 8.4-5
Notation
n = sample size
= sample mean
= population mean
x
x


Section 8.4-6
Requirements
1) The sample is a simple random sample.
2) Either or both of these conditions is satisfied:
The population is normally distributed or n > 30.

Section 8.4-7
Test Statistic
x
x
t
s
n




Section 8.4-8
Running the Test
P-values: Use technology or use the Student t
distribution in Table A-3 with degrees of freedom
df = n – 1.
Critical values: Use the Student t distribution with
degrees of freedom df = n – 1.

Section 8.4-9
Important Properties of the
Student t Distribution
1.The Student t distribution is different for different sample
sizes (see Figure 7-5 in Section 7-3).
2.The Student t distribution has the same general bell shape
as the normal distribution; its wider shape reflects the
greater variability that is expected when s is used to
estimate σ.
3.The Student t distribution has a mean of t = 0.
4.The standard deviation of the Student t distribution varies
with the sample size and is greater than 1.
5.As the sample size n gets larger, the Student t distribution
gets closer to the standard normal distribution.

Section 8.4-10
Example
Listed below are the measured radiation emissions (in W/kg)
corresponding to a sample of cell phones.
Use a 0.05 level of significance to test the claim that cell
phones have a mean radiation level that is less than 1.00
W/kg.
The summary statistics are: .
0.38 0.55 1.54 1.55 0.50 0.60 0.92 0.96 1.00 0.86 1.46
0.938 and 0.423
x s
 

Section 8.4-11
Example - Continued
Requirement Check:
1. We assume the sample is a simple random sample.
2. The sample size is n = 11, which is not greater than 30, so
we must check a normal quantile plot for normality.

Section 8.4-12
Example - Continued
The points are reasonably close to a straight line and there is
no other pattern, so we conclude the data appear to be from a
normally distributed population.

Section 8.4-13
Example - Continued
Step 1: The claim that cell phones have a mean radiation
level less than 1.00 W/kg is expressed as μ < 1.00 W/kg.
Step 2: The alternative to the original claim is μ ≥ 1.00 W/kg.
Step 3: The hypotheses are written as:
0
1
: 1.00 W/kg
: 1.00 W/kg
H
H





Section 8.4-14
Example - Continued
Step 4: The stated level of significance is α = 0.05.
Step 5: Because the claim is about a population mean μ, the
statistic most relevant to this test is the sample mean: .
x

Section 8.4-15
Example - Continued
Step 6: Calculate the test statistic and then find the P-value or
the critical value from Table A-3:
0.938 1.00
0.486
0.423
11
x
x
t
s
n

 
   

Section 8.4-16
Example - Continued
Step 7: Critical Value Method: Because the test statistic of
t = –0.486 does not fall in the critical region bounded by the
critical value of t = –1.812, fail to reject the null hypothesis.

Section 8.4-17
Example - Continued
Step 7: P-value method: Technology, such as a TI-83/84 Plus
calculator can output the P-value of 0.3191. Since the P-value
exceeds α = 0.05, we fail to reject the null hypothesis.

Section 8.4-18
Example
Step 8: Because we fail to reject the null hypothesis, we
conclude that there is not sufficient evidence to support the
claim that cell phones have a mean radiation level that is less
than 1.00 W/kg.

Section 8.4-19
a) In a left-tailed hypothesis test, the sample size is n = 12,
and the test statistic is t = –2.007.
b) In a right-tailed hypothesis test, the sample size is n = 12,
and the test statistic is t = 1.222.
c) In a two-tailed hypothesis test, the sample size is n = 12,
and the test statistic is t = –3.456.
Assuming that neither software nor a TI-83 Plus calculator is
available, use Table A-3 to find a range of values for the P-
value corresponding to the given results.
Finding P-Values

Section 8.4-20
Example – Confidence
Interval Method
We can use a confidence interval for testing a claim about μ.
For a two-tailed test with a 0.05 significance level, we
construct a 95% confidence interval.
For a one-tailed test with a 0.05 significance level, we
construct a 90% confidence interval.

Section 8.4-21
Interval Method
Using the cell phone example, construct a confidence interval
that can be used to test the claim that μ < 1.00 W/kg,
assuming a 0.05 significance level.
Note that a left-tailed hypothesis test with α = 0.05
corresponds to a 90% confidence interval.
Using methods described in Section 7.3, we find:
0.707 W/kg < μ < 1.169 W/kg

Section 8.4-22
Interval Method
Because the value of μ = 1.00 W/kg is contained in the
interval, we cannot reject the null hypothesis that μ = 1.00
W/kg .
Based on the sample of 11 values, we do not have sufficient
evidence to support the claim that the mean radiation level is
less than 1.00 W/kg.

Section 8.4-23
Part 2
When σ is known, we use test that involves the standard
normal distribution.
In reality, it is very rare to test a claim about an unknown
population mean while the population standard deviation is
somehow known.
The procedure is essentially the same as a t test, with the
following exception:

Section 8.4-24
Test Statistic for Testing a Claim
About a Mean (with σ Known)
The test statistic is:
The P-value can be provided by technology or the
standard normal distribution (Table A-2).
The critical values can be found using the standard normal
distribution (Table A-2).
x
x
z
n





Section 8.4-25
Example
If we repeat the cell phone radiation example, with the
assumption that σ = 0.480 W/kg, the test statistic is:
The example refers to a left-tailed test, so the P-value is the
area to the left of z = –0.43, which is 0.3336 (found in Table A-
2).
Since the P-value is large, we fail to reject the null and reach
the same conclusion as before.
0.938 1.00
0.43
0.480
11
x
x
z
n


 
   

Chapter 8 Section 4.ppt

More Related Content

Similar to Chapter 8 Section 4.ppt (20)

More from ManoloTaquire (20)

Recently uploaded (20)

Chapter 8 Section 4.ppt