Unformatted text preview: BIT 2405 Quantitative Methods I
BIT 2405 Quantitative Methods I
Chapter 11.1
Inferences About A Population Variance This Week
This Week
Lectures Hawkes Today Optional HW911 and
REQUIRED Quiz 3 are Posted. Next Week and Beyond.
Next Week and Beyond.
Lectures Hawkes
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work on Hawkes
EARLY
EARLY
9.1 is not required
but you may want
but
to do the instruct.
to Email me if you want to take the Early Test 3. Questions?
Questions? Chapter 9 & 11.1
Chapter 9 & 11.1 Hypothesis Tests
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KEY
Lecture #
Chapter TEXT
Quantitative Methods I,
Anderson, Sweeney & Williams Hawkes
Certifications due
on Monday
on
@ 11:59pm ! Hawkes Learning Systems:
Statistics
Certifications Section Topic 11.1 9.2 Hypothesis Testing
Proportions: P Value 9.3 Hypothesis Testing
Proportions: z Value
Hypothesis Testing
Means: P Value
Hypothesis Testing
Means: z Value
Hypothesis Testing
Means: t Value
Hypothesis Testing Inferences about a
Population Variance Today 17a
Ch11.ppt Section Topic 9.4 17b
Hypothesi
sTesting.p
pt 9 & 11 Overview and
Summary of All the
Hypothesis Tests 9.5
9.6
9.9 Chapter 11
Chapter 11 Inferences About Population Variances
11.1 Inference about a Population Variance
s Inferences about the Variances of Two Populations Inferences About a Population Variance
Inferences About a Population Variance
s
s
s ChiSquare Distribution
Interval Estimation of σ
Hypothesis Testing 2 Chi
Chi
is Pronounced
is
‘ kī ChiSquare Distribution
ChiSquare Distribution
s The chisquare distribution is the sum of squared standardized normal random variables such as (z1)2+(z2)2+(z3)2 and so on. The chisquare distribution is based on sampling from a normal population. The sampling distribution of (n 1)s2/σ has a chi square distribution whenever a simple random sample of size n is selected from a normal population.
2 We can use the chisquare distribution to develop interval estimates and conduct hypothesis tests about a population variance. Examples of Sampling Distribution of (n 1)s /σ
Examples of Sampling Distribution of (
2 2 With 2 degrees of freedom
With 5 degrees of freedom
With 10 degrees of freedom 0 (n − 1) s 2
σ2
The ChiSquare Distribution Is NOT SYMMETRICAL
nor Centered on ZERO! Examples of Sampling Distribution of (n 1)s /σ
Examples of Sampling Distribution of (
2 2 With 2 degrees of freedom
With 5 degrees of freedom
With 10 degrees of freedom 0 (n − 1) s 2
σ2 When df > 100, X2 becomes approximately normally distributed (mean = n and variance = 2n). We will NOT have sample sizes > 101 so you can use the X2 tables. ChiSquare Distribution
s s χa
We will use the notation to denote the value for the chisquare distribution that provides an area of a to the 2
χa
right of the stated value.
2 For example, there is a .95 probability of obtaining a χ 2 (chisquare) value such that
2
2
χ.975 ≤ χ 2 ≤ χ.025 0.975  0.025 = 0.950 Interval Estimation of σ
2 2
χ.975 ≤ ( n − 1)s 2 .025 .025 95% of the
possible χ 2 values
0 2
χ.975 σ2 2
≤ χ.025 2
χ.025 χ2 Looking up χ in the Table.
Looking up 2 1. A rea given is the Upper Tail
Area
f or a χ 2 value.
1. CHISQUARE Distribution
CHISQUARE
i s NOT SYMMETRICAL .
NOT Interval Estimation of σ
Interval Estimation of 2 There is a (1 – α) probability of obtaining a χ 2 value such that
2
2
2
s χ (1−α / 2) ≤ χ ≤ χα / 2 s Substituting (n – 1)s2/σ 2 for the χ 2 we get
2
χ (1−α / 2) s (n − 1) s 2
2
≤
≤ χα / 2
σ2 Performing algebraic manipulation we get
( n − 1) s2
( n − 1) s2
≤ σ2 ≤ 2
2
χα /2
χ (1− α / 2)
Pay close attention to which X2 value you put where!
Pay Interval Estimation of σ
Interval Estimation of s 2 Interval Estimate of a Population Variance
( n − 1) s2
( n − 1) s 2
≤ σ2 ≤ 2
2
χα /2
χ (1− α / 2) where the χ 2 values are based on a chisquare
values are based on a chisquare
distribution with n 1 degrees of freedom and
where 1 α is the confidence coefficient. Interval Estimation of σ
Interval Estimation of s 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. (n − 1) s 2
(n − 1) s 2
≤σ ≤
2
2
χα /2
χ (1−α / 2) Interval Estimation of σ
Interval Estimation of s 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. Interval Estimation of σ
Interval Estimation of 2 s Example: Buyer’s Digest (A) We will use the 10 readings below to develop a
95% confidence interval estimate of the population
variance. Thermostat 1 2 3 4 5 6 7 8 9 10
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 Interval Estimation of σ
2
χ 2 ≤ χ 2 ≤ χ.025 2
.975 For n 1 = 10 1 = 9 d.f. and α = .05
Selected Values from the ChiSquare Distribution Table
Area in Upper Tail D egrees
of Freedom 5
6
7
8
9
10 .99
0.554
0.872
1.239
1.647
2.088 .975
0.831
1.237
1.690
2.180
2.700 .95
1.145
1.635
2.167
2.733
3.325 .90
1.610
2.204
2.833
3.490
4.168 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 The
The .10
9.236
10.645
12.017
13.362
14.684 2
χ.975 value .05
11.070
12.592
14.067
15.507
16.919 .025
12.832
14.449
16.013
17.535
19.023 .01
15.086
16.812
18.475
20.090
21.666 Interval Estimation of σ
Interval Estimation of 2
χ 2 ≤ χ 2 ≤ χ.025 2
.975 For n 1 = 10 1 = 9 d.f. and α = .05 2.700 ≤ χ ≤ χ
2 .025 0 2.700 Area in
Upper Tail
= .975 2
.025 χ2 Interval Estimation of σ
2
χ 2 ≤ χ 2 ≤ χ.025 2
.975 For n 1 = 10 1 = 9 d.f. and α = .05
Selected Values from the ChiSquare Distribution Table
Area in Upper Tail D egrees
of Freedom 5
6
7
8
9
10 .99
0.554
0.872
1.239
1.647
2.088 .975
0.831
1.237
1.690
2.180
2.700 .95
1.145
1.635
2.167
2.733
3.325 .90
1.610
2.204
2.833
3.490
4.168 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 The
The .10
9.236
10.645
12.017
13.362
14.684 2
χ.025 value .05
11.070
12.592
14.067
15.507
16.919 .025
12.832
14.449
16.013
17.535
19.023 .01
15.086
16.812
18.475
20.090
21.666 Interval Estimation of σ
2
χ 2 ≤ χ 2 ≤ χ.025 2
.975 n 1 = 10 1 = 9 degrees of freedom and α = .05
2.700 ≤ χ 2 ≤ 19.023 .025 0 2.700 Area in Upper
Tail = .025 19.023 χ2 Interval Estimation of σ
Interval Estimation of s 2 Sample variance s2 provides a point estimate of σ 2.
∑ ( xi − x ) 2 6. 3
s=
=
= . 70
n −1
9
2 s A 95% confidence interval for the population variance is given by:
(10 − 1). 70
(10 − 1). 70
2
≤σ ≤
19. 02
2. 70
2
.33 < σ 2 < 2.33 Formula ( n − 1) s2
( n − 1) s2
≤ σ2 ≤ 2
2
χα /2
χ (1− α / 2) 2.700 ≤ χ 2 ≤ 19.023 Hypothesis Testing
Hypothesis Testing
About a Population Variance
s LowerTail Test
•Hypotheses 2
H 0 : σ 2 ≥ σ 0
2
H a : σ 2 < σ 0 2 σ0
where is the hypothesized value
for the population variance •Test Statistic (n − 1) s 2
2
χ obs =
σ 02 Hypothesis Testing
About a Population Variance
s LowerTail Test (continued)
•Rejection Rule
Critical value approach: 2
2
Reject H0 if χ ≤ χ (1−α )
obs pValue approach: Reject H0 if pvalue < α 2
χ(1−α )
where is based on a chisquare
distribution with n 1 d.f. Hypothesis Testing
About a Population Variance
s UpperTail Test
•Hypotheses H0 : σ 2 ≤ σ 2
0
Ha : σ 2 > σ 2
0
2 σ0
where is the hypothesized value
for the population variance •Test Statistic ( n − 1) s 2
χ2 =
σ2
obs
0 Hypothesis Testing
About a Population Variance
s UpperTail Test (continued)
•Rejection Rule
Critical value approach: Reject H0 if χ 2 ≥ χ α2
obs pValue approach: Reject H0 if pvalue < α 2
χα
where is based on a chisquare
distribution with n 1 d.f. Hypothesis Testing
About a Population Variance
s TwoTail Test
•Hypotheses H0 : σ 2 = σ 2
0
Ha : σ 2 ≠ σ 2
0
2 σ0
where is the hypothesized value
for the population variance •Test Statistic ( n − 1) s 2
χ2 =
σ2
obs
0 Hypothesis Testing
About a Population Variance
s TwoTail Test (continued)
•Rejection Rule
Critical value approach:
2
2
Reject H0 if χ 2 ≤ χ (1−α / 2 ) or χ 2 ≥ χα / 2
obs obs pValue approach:
Reject H0 if pvalue < α
2
χ (1−α / 2) and χ α2 / 2
where are based on a
chisquare distribution with n 1 d.f. Hypothesis Testing
About a Population Variance
s Example: Buyer’s Digest (B) Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s Digest gives an “acceptable”
rating to a thermostat with a temperature variance
of 0.5 or less. We will conduct a hypothesis test (with α = .10)
to determine whether the ThermoRite thermostat’s
temperature variance is “acceptable”. Hypothesis Testing
Hypothesis Testing
About a Population Variance
s Example: Buyer’s Digest (B) Using the 10 readings, we will conduct a
hypothesis test (with α = .10) to determine whether
the ThermoRite thermostat’s temperature variance is
“acceptable”. Thermostat 1 2 3 4 5 6 7 8 9 10 Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 Hypothesis Testing
About a Population Variance
s Hypotheses
H 0 : σ 2 ≤ 0.5 H a : σ 2 > 0.5
s Rejection Rule
Reject H0 if χ 2 > 14.684
obs Upper
tail
test Hypothesis Testing
About a Population Variance For n 1 = 10 1 = 9 d.f. and α = .10
Selected Values from the ChiSquare Distribution Table
Area in Upper Tail D egrees
of Freedom 5
6
7
8
9
10 .99
0.554
0.872
1.239
1.647
2.088 .975
0.831
1.237
1.690
2.180
2.700 .95
1.145
1.635
2.167
2.733
3.325 .90
1.610
2.204
2.833
3.490
4.168 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 The
The 2
χ.10value .10
9.236
10.645
12.017
13.362
14.684 .05
11.070
12.592
14.067
15.507
16.919 .025
12.832
14.449
16.013
17.535
19.023 .01
15.086
16.812
18.475
20.090
21.666 Hypothesis Testing
About a Population Variance
s Rejection Region χ2 =
obs ( n − 1)s 2 σ2 9s 2
=
.5 Area in Upper
Tail = .10 0 14.684 χ2
Reject H0 Hypothesis Testing
About a Population Variance
s Test Statistic
The sample variance s 2 = 0.7
9(.7)
χ=
= 12.6
.5
obs
2 s Conclusion Because χ 2 = 12.6 is less than 14.684, we cannot
obs
reject H0. The sample variance s2 = .7 is insufficient
evidence to conclude that the temperature variance
for ThermoRite thermostats is unacceptable. Hypothesis Testing
About a Population Variance
s Using the pValue
• The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate pvalue is less than .90 (χ 2 = 4.168) and greater than .10 (χ 2 = 14.684).
• Because the p –value > α = .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). Hypothesis Testing
About a Population Variance
s 9(.7)
Using the pValue
2
χ obs =
= 12.6
• The rejection region for the ThermoRite .5 thermostat example is in the upper tail; thus, the appropriate pvalue is less than .90 (χ 2 = 4.168) and greater than .10 (χ 2 = 14.684).
• Because the p –value > α = .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). Hypothesis Testing
About a Population Variance
s Using the pValue
• The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate pvalue is less than .90 (χ 2 = 4.168) and greater than .10 (χ 2 = 14.684).
• Because the p –value > α = .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). Recap of Hypothesis Testing
Recap of Hypothesis Testing ...
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 Summer '03
 DHBannan
 ChiSquare Test, Variance, Statistical hypothesis testing, Fisher's method, population variance, Upper Tail

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