John Loucks St . Edward’s University
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Transcript of John Loucks St . Edward’s University
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John LoucksSt. Edward’sUniversity
...........
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Chapter 11 Inferences About Population Variances
Inference about a Population Variance Inferences about Two Populations Variances
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Inferences About a Population Variance
If the sample variance is excessive, overfilling and underfilling may be occurring even though the mean is correct.
The mean filling weight is important, but also is the
variance of the filling weights.
Consider the production process of filling containers with a liquid detergent product.
A variance can provide important decision-making
information.
By selecting a sample of containers, we can compute
a sample variance for the amount of detergent placed in a container.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Inferences About a Population Variance
Chi-Square Distribution Interval Estimation of 2
Hypothesis Testing
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Chi-Square Distribution
We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance.
The sampling distribution of (n - 1)s2/2 has a chi-
square distribution whenever a simple random sample of size n is selected from a normal population.
The chi-square distribution is based on sampling from a normal population.
The chi-square distribution is the sum of squared
standardized normal random variables such as (z1)2+(z2)2+(z3)2 and so on.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Examples of Sampling Distribution of (n - 1)s2/2
0
With 2 degrees of freedom
2
2
( 1)n s
With 5 degrees of freedom
With 10 degrees of freedom
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or duplicated, or posted to a publicly accessible website, in whole or in part.
2 2 2.975 .025
Chi-Square Distribution
For example, there is a .95 probability of obtaining a 2 (chi-square) value such that
We will use the notation to denote the value for the chi-square distribution that provides an area of a to the right of the stated value.
2a
2a
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or duplicated, or posted to a publicly accessible website, in whole or in part.
95% of thepossible 2 values
2
0
.025
2.025
.025
2.975
Interval Estimation of 2
22 2.975 .0252
( 1)n s
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of 2
( ) ( )
/ ( / )
n s n s
1 12
22
22
1 22
a a
( ) ( )
/ ( / )
n s n s
1 12
22
22
1 22
a a
2 2 2(1 / 2) / 2a a
22 2(1 / 2) / 22
( 1)n sa a
Substituting (n – 1)s2/2 for the 2 we get
Performing algebraic manipulation we get
There is a (1 – a) probability of obtaining a 2 value
such that
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimate of a Population Variance
Interval Estimation of 2
( ) ( )
/ ( / )
n s n s
1 12
22
22
1 22
a a
( ) ( )
/ ( / )
n s n s
1 12
22
22
1 22
a a
where the values are based on a chi-squaredistribution with n - 1 degrees of freedom andwhere 1 - a is the confidence coefficient.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of
Interval Estimate of a Population Standard Deviation Taking the square root of the upper and lower
limits of the variance interval provides the confidenceinterval for the population standard deviation.
2 2
2 2/ 2 (1 / 2)
( 1) ( 1)n s n s
a a
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Buyer’s Digest rates thermostats manufactured for
home temperature control. In a recent test, 10thermostats manufactured by ThermoRite
wereselected and placed in a test room that wasmaintained at a temperature of 68oF. Thetemperature readings of the ten thermostats
areshown on the next slide.
Interval Estimation of 2
Example: Buyer’s Digest (A)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Interval Estimation of 2
We will use the 10 readings below to develop a95% confidence interval estimate of the populationvariance.
Example: Buyer’s Digest (A)
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2Thermostat 1 2 3 4 5 6 7 8 9 10
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Degreesof Freedom .99 .975 .95 .90 .10 .05 .025 .01
5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666
10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209
Area in Upper Tail
Interval Estimation of 2
Selected Values from the Chi-Square Distribution Table
Our value
2.975
For n - 1 = 10 - 1 = 9 d.f. and a = .05
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Interval Estimation of 2
2
0
.025
2
2.0252
( 1)2.700 n s
Area inUpper Tail
= .975
2.700
For n - 1 = 10 - 1 = 9 d.f. and a = .05
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Degreesof Freedom .99 .975 .95 .90 .10 .05 .025 .01
5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666
10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209
Area in Upper Tail
Interval Estimation of 2
Selected Values from the Chi-Square Distribution Table
For n - 1 = 10 - 1 = 9 d.f. and a = .05
Our value 2
.025
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or duplicated, or posted to a publicly accessible website, in whole or in part.
2
0
.025
2.700
Interval Estimation of 2
n - 1 = 10 - 1 = 9 degrees of freedom and a = .05
2
2( 1)2.700 19.023n s
19.023
Area in UpperTail = .025
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Sample variance s2 provides a point estimate of 2.
s x xni2
2
16 39
70
( ) . .s x xni2
2
16 39
70
( ) . .
( )..
( )..
10 1 7019 02
10 1 702 70
2 ( ).
.( ).
.10 1 70
19 0210 1 70
2 702
Interval Estimation of 2
.33 < 2 < 2.33
A 95% confidence interval for the population variance is given by:
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Left-Tailed Test
Hypothesis TestingAbout a Population Variance
22
021 ( )n s
2
2
021 ( )n s
where is the hypothesized valuefor the population variance
20
• Test Statistic
• Hypotheses 2 20 0: H
2 20: aH
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Left-Tailed Test (continued)
Hypothesis TestingAbout a Population Variance
Reject H0 if p-value < ap-Value approach:
Critical value approach:• Rejection Rule
Reject H0 if 2 2(1 )a
where is based on a chi-squaredistribution with n - 1 d.f.
2(1 )a
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Right-Tailed Test
Hypothesis TestingAbout a Population Variance
H02
02: H0
202:
Ha : 202Ha : 202
22
021 ( )n s
2
2
021 ( )n s
where is the hypothesized valuefor the population variance
20
• Test Statistic
• Hypotheses
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Right-Tailed Test (continued)
Hypothesis TestingAbout a Population Variance
Reject H0 if 2 2a
Reject H0 if p-value < a
2awhere is based on a chi-square
distribution with n - 1 d.f.
p-Value approach:
Critical value approach:• Rejection Rule
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Two-Tailed Test
Hypothesis TestingAbout a Population Variance
22
021 ( )n s
2
2
021 ( )n s
where is the hypothesized valuefor the population variance
20
• Test Statistic
• Hypotheses
Ha : 202Ha : 202
H02
02: H0
202:
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Two-Tailed Test (continued)
Hypothesis TestingAbout a Population Variance
Reject H0 if p-value < ap-Value approach:
Critical value approach:• Rejection Rule
2 2 2 2(1 / 2) / 2 or a a Reject H0 if
where are based on achi-square distribution with n - 1 d.f.
2 2(1 / 2) / 2 and a a
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Recall that Buyer’s Digest is rating ThermoRitethermostats. Buyer’s Digest gives an “acceptable”rating to a thermostat with a temperature varianceof 0.5 or less.
Hypothesis TestingAbout a Population Variance
Example: Buyer’s Digest (B)
We will conduct a hypothesis test (with a = .10)to determine whether the ThermoRite thermostat’stemperature variance is “acceptable”.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis TestingAbout a Population Variance
Using the 10 readings, we will conduct ahypothesis test (with a = .10) to determine whetherthe ThermoRite thermostat’s temperature variance is“acceptable”.
Example: Buyer’s Digest (B)
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2Thermostat 1 2 3 4 5 6 7 8 9 10
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypotheses2
0 : 0.5H 2: 0.5aH
Hypothesis TestingAbout a Population Variance
Reject H0 if 2 > 14.684
Rejection Rule
Right-tailedtest
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Degreesof Freedom .99 .975 .95 .90 .10 .05 .025 .01
5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666
10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209
Area in Upper TailSelected Values from the Chi-Square Distribution Table
For n - 1 = 10 - 1 = 9 d.f. and a = .10
Hypothesis TestingAbout a Population Variance
Our value 2
.10
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or duplicated, or posted to a publicly accessible website, in whole or in part.
2
0 14.684
Area in UpperTail = .10
Hypothesis TestingAbout a Population Variance
Rejection Region
2 22
2( 1) 9
.5n s s
Reject H0
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Test Statistic
2 9(.7) 12.6.5
Hypothesis TestingAbout a Population Variance
Because 2 = 12.6 is less than 14.684, we cannotreject H0. The sample variance s2 = .7 is insufficientevidence to conclude that the temperature variancefor ThermoRite thermostats is unacceptable.
Conclusion
The sample variance s 2 = 0.7
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Using the p-Value
• The sample variance of s 2 = .7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5).
• Because the p –value > a = .10, we cannot reject the null hypothesis.
• The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p-value is less than .90 (2 = 4.168) and greater than .10 (2 = 14.684).
Hypothesis TestingAbout a Population Variance
The exact p-value is .18156.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Inferences About Two Population Variances
The two sample variances will be the basis for making
inferences about the two population variances.
We use data collected from two independent random sample, one from population 1 and another from population 2.
We may want to compare the variances in: product quality resulting from two different
production processes,
assembly times for two assembly methods.
temperatures for two heating devices, or
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or duplicated, or posted to a publicly accessible website, in whole or in part.
One-Tailed Test
• Test Statistic
• Hypotheses
Hypothesis Testing About theVariances of Two Populations
Denote the population providing thelarger sample variance as population 1.
2 20 1 2: H
2 21 2: aH
21 2
2
sF s
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or duplicated, or posted to a publicly accessible website, in whole or in part.
One-Tailed Test (continued)
Reject H0 if p-value < a
where the value of Fa is based on anF distribution with n1 - 1 (numerator)and n2 - 1 (denominator) d.f.
p-Value approach:
Critical value approach:• Rejection Rule
Hypothesis Testing About theVariances of Two Populations
Reject H0 if F > Fa
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Two-Tailed Test
• Test Statistic
• Hypotheses
Hypothesis Testing About theVariances of Two Populations
H0 12
22: H0 1
222:
Ha : 12
22Ha : 1
222
Denote the population providing thelarger sample variance as population 1.
21 2
2
sF s
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Two-Tailed Test (continued)
Reject H0 if p-value < ap-Value approach:
Critical value approach:• Rejection Rule
Hypothesis Testing About theVariances of Two Populations
Reject H0 if F > Fa/2
where the value of Fa/2 is based on anF distribution with n1 - 1 (numerator)and n2 - 1 (denominator) d.f.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Buyer’s Digest has conducted the same test, as was
described earlier, on another 10 thermostats, this time
manufactured by TempKing. The temperature readings
of the ten thermostats are listed on the next slide.
Hypothesis Testing About theVariances of Two Populations
Example: Buyer’s Digest (C)
We will conduct a hypothesis test with a = .10 to seeif the variances are equal for ThermoRite’s thermostatsand TempKing’s thermostats.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Hypothesis Testing About theVariances of Two Populations
Example: Buyer’s Digest (C)ThermoRite Sample
TempKing Sample
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2Thermostat 1 2 3 4 5 6 7 8 9 10
Temperature 67.7 66.4 69.2 70.1 69.5 69.7 68.1 66.6 67.3 67.5Thermostat 1 2 3 4 5 6 7 8 9 10
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or duplicated, or posted to a publicly accessible website, in whole or in part.
HypothesesH0 1
222: H0 1
222:
Ha : 12
22Ha : 1
222
Hypothesis Testing About theVariances of Two Populations
Reject H0 if F > 3.18
The F distribution table (on next slide) shows that withwith a = .10, 9 d.f. (numerator), and 9 d.f. (denominator),F.05 = 3.18.
(Their variances are not equal)
(TempKing and ThermoRite thermostatshave the same temperature variance)
Rejection Rule
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Denominator Area inDegrees Upper
of Freedom Tail 7 8 9 10 158 .10 2.62 2.59 2.56 2.54 2.46
.05 3.50 3.44 3.39 3.35 3.22.025 4.53 4.43 4.36 4.30 4.10.01 6.18 6.03 5.91 5.81 5.52
9 .10 2.51 2.47 2.44 2.42 2.34.05 3.29 3.23 3.18 3.14 3.01
.025 4.20 4.10 4.03 3.96 3.77.01 5.61 5.47 5.35 5.26 4.96
Numerator Degrees of FreedomSelected Values from the F Distribution Table
Hypothesis Testing About theVariances of Two Populations
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Test Statistic
Hypothesis Testing About theVariances of Two Populations
We cannot reject H0. F = 2.53 < F.05 = 3.18.There is insufficient evidence to conclude thatthe population variances differ for the twothermostat brands.
Conclusion
21 2
2
sF s = 1.768/.700 = 2.53
TempKing’s sample variance is 1.768ThermoRite’s sample variance is .700
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Determining and Using the p-Value
Hypothesis Testing About theVariances of Two Populations
• Because a = .10, we have p-value > a and therefore we cannot reject the null hypothesis.
• But this is a two-tailed test; after doubling the upper-tail area, the p-value is between .20 and .10.
• Because F = 2.53 is between 2.44 and 3.18, the area in the upper tail of the distribution is between .10 and .05.
Area in Upper Tail .10 .05 .025 .01F Value (df1 = 9, df2 = 9) 2.44 3.18 4.03 5.35
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End of Chapter 11