Section 8-5
Testing a Claim About a
Mean:  Not Known
1
Notation
n = sample size
x = sample mean
m = claimed population mean (from H0)
s = sample standard deviation
2
Requirements for Testing Claims
About a Population
Mean (with  Not Known)
1) The value of the population standard
deviation  is not known.
2) Either or both of these conditions is
satisfied: The population is normally
distributed or n > 30.
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Test Statistic for Testing a
Claim About a Mean
(with  Not Known)
x–µ
t= s
n
P-values and Critical Values
Found in Table A-3 or by calculator
Degrees of freedom (df) = n – 1
4
Example:
People have died in boat accidents because an
obsolete estimate of the mean weight of men
(166.3 lb) was used.
A random sample of n = 40 men yielded the mean
x = 172.55 lb and standard deviation s = 26.33 lb.
Do not assume that the population standard
deviation  is known.
Test the claim that men have a mean weight
greater than 166.3 lb.
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Example:
Requirements are satisfied:  is not
known, sample size is 40 (n > 30)
We can express claim as m > 166.3 lb
It does not contain equality, so it is the
alternative hypothesis.
H0: m = 166.3 lb null hypothesis
H1: m > 166.3 lb alternative hypothesis
(and original claim)
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Example:
Let us set significance level to  = 0.05
Next we calculate t
t 
x  mx
s
n

172.55  166.3
26.33
 1.501
40
df = n – 1 = 39
area of 0.05, one-tail yields
critical value t = 1.685;
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Example:
t = 1.501 does not fall in the critical
region bounded by t = 1.685, we fail
to reject the null hypothesis.
m = 166.3
or
z=0
x  172.55
Critical value
t = 1.685
or
t = 1.52
8
Example:
Final conclusion:
Because we fail to reject the null hypothesis, we
conclude that there is not sufficient evidence to
support a conclusion that the population mean
is greater than 166.3 lb.
9
Normal Distribution Versus
Student t Distribution
The critical value in the preceding example
was t = 1.782, but if the normal distribution
were being used, the critical value would have
been z = 1.645.
The Student t critical value is larger (farther to
the right), showing that with the Student t
distribution, the sample evidence must be
more extreme before we can consider it to be
significant.
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P-Value Method
 Use software or a TI-83/84 Plus
calculator.
 If technology is not available, use Table
A-3 to identify a range of P-values
(this will be explained in Section 8.6)
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Testing hypothesis by TI-83/84
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•
•
•
•
•
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Press STAT and select TESTS
Scroll down to T-Test press ENTER
Choose Data or Stats. For Stats:
Type in m0: (claimed mean, from H0)
x: (sample mean)
sx: (sample st. deviation)
n: (sample size)
choose H1: m ≠m0
<m0
>m0
(two tails) (left tail) (right tail)
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• (continued)
• Press on Calculate
• Read the test statistic t=…
• and the P-value p=…
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Section 8-6
Testing a Claim About a
Standard Deviation or
Variance
14
Notation
n = sample size
s = sample standard deviation
s2 = sample variance
 = claimed value of the population standard
deviation (from H0 )
2 = claimed value of the population
variance (from H0 )
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Requirements for Testing
Claims About  or  2
1. The sample is a simple random
sample.
2. The population has a normal
distribution. (This is a much stricter
requirement than the requirement of a
normal distribution when testing
claims about means.)
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Chi-Square Distribution
Test Statistic
 
2
n  1s

2
2
17
Critical Values for
Chi-Square Distribution
• Use Table A-4.
• The degrees of freedom df = n –1.
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Table A-4
Table A-4 is based on cumulative areas
from the right.
Critical values are found in Table A-4 by
first locating the row corresponding to the
appropriate number of degrees of freedom
(where df = n –1).
Next, the significance level  is used to
determine the correct column.
The following examples are based on a
significance level of  = 0.05.
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Critical values
Right-tailed test: needs one critical value
Because the area to the right of the
critical value is 0.05, locate 0.05 at the
top of Table A-4.
Area 
Area 1- 


cr itical value
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Critical values
Left-tailed test: needs one critical value
With a left-tailed area of 0.05, the area to
the right of the critical value is 0.95, so
locate 0.95 at the top of Table A-4.
Area 
Area 1- 
 1 - 
cr itical value
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Critical values
Two-tailed test: needs two critical values
Critical values are two different positive
numbers, both taken from Table A-4
Divide a significance level of 0.05 between
the left and right tails, so the areas to the
right of the two critical values are 0.975
and 0.025, respectively.
Locate 0.975 and 0.025 at top of Table A-4
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Critical values for a two-tailed test
Area 
Area 
Area 1- 

  
left critical value

 
right critical value
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Finding a range for P-value
• A useful interpretation of the P-value: it is
observed level of significance.
• Compare your test statistic 2 with critical
values shown in Table A-4 on the line with
df=n-1 degrees of freedom.
• Find the two critical values that enclose your
test statistic. Determine the significance
levels 1 and 2 for those two critical values.
• Your P-value is between 1 and 2
(see examples below)
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Examples:
Right-tailed test: If the test statistic 2 is
between critical values corresponding to
the areas 1 and 2 , then your P-value
is between 1 and 2 .
Left-tailed test: If the test statistic 2 is
between critical values corresponding to
the areas 1-1 and 1-2 , then your
P-value is between 1 and 2 .
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Examples:
Two-tailed test: If the test statistic 2 is
between critical values corresponding to
the areas 1 and 2 , then your P-value
is between 21 and 22 .
Two-tailed test: If the test statistic 2 is
between critical values corresponding to
the areas 1-1 and 1-2 , then your
P-value is between 21 and 22 .
(Note: for two-tailed tests, multiply the
areas by two)
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Finding the exact P-value by TI-83/84
• Use the test statistic 2 and the calculator function
2 cdf to compute the area of the tail:
2 cdf(teststat,999,df) gives the area of the right tail
(to the right from the test statistic)
2 cdf(-999,teststat,df) gives the area of the left tail
(to the left from the test statistic)
Multiply the area of the tail by 2
if you have a two-tailed test
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