1. Chapter 11 Inferences About Population Variances
Inference about a Population Variance
Inferences about Two Population Variances

2. Inferences About a Population Variance
Chi-Square Distribution
Interval Estimation
Hypothesis Testing
卡方分布

3. Chi-Square Distribution
The chi-square distribution is the sum of squared
standardized normal random variables such as
(z1)2+(z2)2+(z3)2 and so on.
The chi-square distribution is based on sampling
from a normal population.
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.
卡方分布 (χ2分布)是概率论与统计学中常用的一种概率分布。k 个独立的标准正态分布变量的平方和服从自由度为k 的卡方分布。卡方分布常用于假设检验和置信区间的计算。
We can use the chi-square distribution to develop
interval estimates and conduct hypothesis tests
about a population variance.

4. Examples of Sampling Distribution of (n - 1)s2/ 2
With 2 degrees
of freedom
With 5 degrees
of freedom
With 10 degrees
of freedom

6. Chi-Square Distribution
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.
For example, there is a .95 probability of obtaining a c2 (chi-square) value such that

7. Interval Estimation of 2
.025
.025
95% of the
possible 2 values 2

8. Interval Estimation of 2
There is a (1 – a) probability of obtaining a c2 value
such that
Substituting (n – 1)s2/s 2 for the c2 we get
Performing algebraic manipulation we get

9. Interval Estimation of 2
Interval Estimate of a Population Variance
where the values are based on a chi-square
distribution with n - 1 degrees of freedom and
where 1 -  is the confidence coefficient.

10. 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 confidence
interval for the population standard deviation.

11. Interval Estimation of 2
Example: Buyer’s Digest (A)
Buyer’s Digest rates thermostats
manufactured for home temperature
control. In a recent test, 10 thermostats
manufactured by ThermoRite were
selected and placed in a test room that
was maintained at a temperature of 68oF.
The temperature readings of the ten thermostats are
shown on the next slide.

12. Interval Estimation of 2
Example: Buyer’s Digest (A)
We will use the 10 readings below to
develop a 95% confidence interval
estimate of the population variance.
Thermostat
Temperature

13. Interval Estimation of 2
For n - 1 = = 9 d.f. and a = .05
Selected Values from the Chi-Square Distribution Table
Our value

14. Interval Estimation of 2
For n - 1 = = 9 d.f. and a = .05
.025
Area in
Upper Tail
= .975 2
2.700

15. Interval Estimation of 2
For n - 1 = = 9 d.f. and a = .05
Selected Values from the Chi-Square Distribution Table
Our value

16. Interval Estimation of 2
n - 1 = = 9 degrees of freedom and a = .05
.025
Area in Upper
Tail = .025 2
2.700
19.023

17. Interval Estimation of 2
Sample variance s2 provides a point estimate of  2.
A 95% confidence interval for the population variance is given by:
.33 < 2 < 2.33

18. Hypothesis Testing About a Population Variance
Lower-tail test:
H0: σ2  σ02
H1: σ2 < σ02
Upper-tail test:
H0: σ2 ≤ σ02
H1: σ2 > σ02
Two-tail test:
H0: σ2 = σ02
H1: σ2 ≠ σ02 a a
a/2
a/2
Reject H0 if
Reject H0 if
Reject H0 if or

19. Hypothesis Testing About a Population Variance
Example: Buyer’s Digest (B)
Recall that Buyer’s Digest is rating
ThermoRite thermostats. Buyer’s Digest
gives an “acceptable” rating to a thermo-
stat with a temperature variance of 0.5
or less.
We will conduct a hypothesis test (with
a = .10) to determine whether the ThermoRite
thermostat’s temperature variance is “acceptable”.

20. Hypothesis Testing About a Population Variance
Example: Buyer’s Digest (B)
Using the 10 readings, we will
conduct a hypothesis test (with a = .10)
to determine whether the ThermoRite
thermostat’s temperature variance is
“acceptable”.
Thermostat
Temperature

21. Hypothesis Testing About a Population Variance
Hypotheses
Rejection Rule
Reject H0 if c 2 >

22. Hypothesis Testing About a Population Variance
For n - 1 = = 9 d.f. and a = .10
Selected Values from the Chi-Square Distribution Table
Our value

23. Hypothesis Testing About a Population Variance
Rejection Region
Area in Upper
Tail = .10 2
14.684
Reject H0

24. Hypothesis Testing About a Population Variance
Test Statistic
The sample variance s 2 = 0.7
Conclusion
Because c2 = 12.6 is less than , we cannot
reject H0. The sample variance s2 = .7 is insufficient
evidence to conclude that the temperature variance
for ThermoRite thermostats is unacceptable.

25. Hypothesis Testing About a Population Variance
Using the p-Value
The rejection region for the ThermoRite
thermostat example is in the upper tail; thus, the
appropriate p-value is less than .90 (c 2 = 4.168)
and greater than .10 (c 2 = ).
A precise p-value
can be found using
Minitab or Excel.
Because the p –value > a = .10, we
cannot reject the null hypothesis.
The sample variance of s 2 = .7 is
insufficient evidence to conclude that the
temperature variance is unacceptable (>.5).

28. Hypothesis Tests for Two Variances
Population
Variances
Goal: Test hypotheses about two
population variances
H0: σx2  σy2
H1: σx2 < σy2
Lower-tail test
F test statistic
H0: σx2 ≤ σy2
H1: σx2 > σy2
Upper-tail test
在统计学的一些应用中，我们或许想比较两种生产线生产出来的产品质量的方差、两种装配方法所需装配时间的方差或者两种加热装置的温度的方差。对两个总体的方差进行比较时，我们可以使用两个独立的随机抽取的样本，设它们分别取自总体1和总体2。两个样本方法可以作为推断两个总体方差的基础。
F分布表是以两个正态分布的抽样分布为基础的。
H0: σx2 = σy2
H1: σx2 ≠ σy2
Two-tail test
The two populations are assumed to be independent and normally distributed

29. F分布
F分布定义为:设X、Y为两个独立的随机变量，X服从自由度为m的卡方分布，Y服从自由度为n的卡方分布，这2 个独立的卡方分布被各自的自由度除以后的比率这一统计量的分布即： F=（x/m）/(y/n) 服从自由度为（m,n)的F-分布， 上式F服从第一自由度为m，第二自由度为n的F分布

30. F分布
F分布的性质
1、它是一种非对称分布；
2、它有两个自由度，即n1 -1和n2-1，相应的分布记为F（ n1 –1， n2-1）， n1 –1通常称为分子自由度， n2-1通常称为分母自由度；
3、F分布是一个以自由度n1 –1和n2-1为参数的分布族，不同的自由度决定了F 分布的形状。
4、F分布的倒数性质：Fα,df1,df2=1/F1-α,df2,df1

31. F分布
F分布表是以两个正态分布的抽样为基础的。F分布不对称，而且F值永远不取负值。任何F分布的形状取决于分子分母的自由度大小。

32. Hypothesis Tests for Two Variances
(continued)
The random variable
Tests for Two
Population
Variances
F test statistic
Has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom
Denote an F value with 1 numerator and 2 denominator degrees of freedom by

33. Test Statistic Tests for Two
Population
Variances
The critical value for a hypothesis test about two population variances is
F test statistic
where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom

34. Hypothesis Testing About the Variances of Two Populations
One-Tailed Test
Hypotheses
Denote the population providing the
larger sample variance as population 1.
Test Statistic

35. Hypothesis Testing About the Variances of Two Populations
One-Tailed Test (continued)
Rejection Rule
Critical value approach:
Reject H0 if F > F
where the value of F is based on an
F distribution with n1 - 1 (numerator)
and n2 - 1 (denominator) d.f.
p-Value approach:
Reject H0 if p-value < a

36. Hypothesis Testing About the Variances of Two Populations
Two-Tailed Test
Hypotheses
Denote the population providing the
larger sample variance as population 1.
Test Statistic

37. 虽然我们可以通过以上公式求左侧F值，但通常在进行假设检验计算时，只需知道右侧F值。进行 假设检验时，可以随意规定哪个总体是总体1或总体2，我们用总体1来代表方差较大的总体时，只有在右侧才有可能出现H0被拒绝的情况。虽然左侧临界值仍存在，但我们不需要知道它值，因为使用方差较大的总体作为总体1，s12/s22往往出现在右侧。

38. Hypothesis Testing About the Variances of Two Populations
Two-Tailed Test (continued)
Rejection Rule
Critical value approach:
Reject H0 if F > F/2
where the value of F/2 is based on an
F distribution with n1 - 1 (numerator)
and n2 - 1 (denominator) d.f.
p-Value approach:
Reject H0 if p-value < a

39. Decision Rules: Two Variances
Use sx2 to denote the larger variance.
H0: σx2 = σy2
H1: σx2 ≠ σy2
H0: σx2 ≤ σy2
H1: σx2 > σy2
/2  F F
Do not
reject H0
Reject H0
Do not
reject H0
Reject H0
rejection region for a two-tail test is:
where sx2 is the larger of the two sample variances

40. Hypothesis Testing About the Variances of Two Populations
Example: Buyer’s Digest (C)
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.
We will conduct a hypothesis test with  = .10 to see
if the variances are equal for ThermoRite’s thermostats
and TempKing’s thermostats.

41. Hypothesis Testing About the Variances of Two Populations
Example: Buyer’s Digest (C)
ThermoRite Sample
Thermostat
Temperature
TempKing Sample
Thermostat
Temperature

42. Hypothesis Testing About the Variances of Two Populations
Hypotheses
(TempKing and ThermoRite thermostats
have the same temperature variance)
(Their variances are not equal)
Rejection Rule
The F distribution table (on next slide) shows that with
with  = .10, 9 d.f. (numerator), and 9 d.f. (denominator),
F.05 = 3.18.
Reject H0 if F > 3.18

43. Hypothesis Testing About the Variances of Two Populations
Selected Values from the F Distribution Table

44. Hypothesis Testing About the Variances of Two Populations
Test Statistic
TempKing’s sample variance is 1.768
ThermoRite’s sample variance is .700
= 1.768/.700 = 2.53
Conclusion
We cannot reject H0. F = 2.53 < F.05 = 3.18.
There is insufficient evidence to conclude that
the population variances differ for the two
thermostat brands.

45. Hypothesis Testing About the Variances of Two Populations
Determining and Using the p-Value
Area in Upper Tail
F Value (df1 = 9, df2 = 9)
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.
But this is a two-tailed test; after doubling the upper-
tail area, the p-value is between .20 and .10. (A precise
p-value can be found using Minitab or Excel.)
Because a = .10, we have p-value > a and therefore
we cannot reject the null hypothesis.

48. End of Chapter 11