Transcript Ch8b
Section 8-5 Testing a Claim About a Mean:
Not Known
1
Notation
n = sample size
x
= sample mean
m
= claimed population mean (from H 0 ) 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 .
3
Test Statistic for Testing a Claim About a Mean (with
Not Known)
t =
x n
– µ
s
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.
5
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.
H
0 :
m
H
1 :
m
= 166.3 lb null hypothesis > 166.3 lb alternative hypothesis (and original claim) 6
Example: Let us set significance level to
= 0.05
Next we calculate t
t
x
m
x s
172.55
166.3
26.33
1.501
n
40
df = n – 1 = 39 area of 0.05, one-tail yields critical value t = 1.685; 7
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
or t =
1.52
Critical value t = 1.685
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.
10
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) 11
Testing hypothesis by TI-83/84
• • • • • • • • Press
STAT
and select Scroll down to
T-Test TESTS
press
ENTER
Choose
Data
or
Stats.
For
Stats:
Type in m
0
: (claimed mean, from H 0 )
x:
(sample mean)
s x :
(sample st. deviation)
n: choose H 1 :
(sample size) m ≠ m 0 < m 0 > m 0 (two tails) (left tail) (right tail)
12
• • • • (continued) Press on
Calculate
Read the test statistic
t=…
and the
P
-value
p=… 13
Section 8-6 Testing a Claim About a Standard Deviation or Variance
14
Notation
n
= sample size
s
= sample standard deviation
s
2 = sample variance
= claimed value of the population standard deviation (from H 0 )
2 = claimed value of the population variance (from H 0 ) 15
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.) 16
Chi-Square Distribution
Test Statistic
2
s
2 2
17
Critical Values for Chi-Square Distribution
•
Use Table A-4.
•
The degrees of freedom df = n –1.
18
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.
19
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 1 critical value Area
20
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 1 critical value Area 1
21
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 22
Critical values for a two-tailed test
Area Area 1 left critic al value Area right critic al value
23
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 (see examples below)
1 and
2 24
Examples:
Right-tailed test : If the test statistic
2 is between critical values corresponding to the areas
1 is between
1 and
2 and
, then your P-value 2 .
Left-tailed test : If the test statistic
2 is between critical values corresponding to the areas 1-
1 and 1-
2 P-value is between
1 and , then your
2 .
25
Examples:
Two-tailed test : If the test statistic
2 is between critical values corresponding to the areas
1 and
2 is between 2
1 , then your P-value and 2
2 .
Two-tailed test : If the test statistic
2 is between critical values corresponding to the areas 1-
1 and 1-
2 P-value is between 2
1 , then your and 2
2 .
(Note: for two-tailed tests, multiply the areas by two) 26
Finding the exact P-value by TI-83/84
• Use the test statistic 2
2 cdf
and the calculator function 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
27