11.2 Comparing Two Standard Deviations

Comparing Standard Deviations from Two Populations
May 05, 2014
11.2 Inferences for σ's, 2 Populations
Study Ch. 11.2, #51, 63­69, 73 Class Notes: Prof. G. Battaly, Westchester Community College, NY
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11.2 Inferences for σ's, 2 Populations
Procedures that assume = σ's
1. Pooled­t test
2. Regression Analysis
3. ANOVA
Previous Techniques to Check for = σ's
1. Box Plots: visually compare spread of data
2. Residual Plot: visually look for non­random pattern that suggests different distances from x­axis
3. Compare s for each population. If any s is 2 or more times any other s, assume that σ's are different.
If the populations really had identical standard deviations, what is the chance of observing as large a discrepancy among sample standard deviations as occurred in the data (or an even larger discrepancy)? Class Notes: Prof. G. Battaly, Westchester Community College, NY
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©Gertrude Battaly, 2014
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1
Comparing Standard Deviations from Two Populations
May 05, 2014
11.2 Inferences for σ's, 2 Populations
Need a more analytical approach
If the populations have identical standard deviations, what is the chance of observing as large a discrepancy among sample standard deviations as occurs in the data? Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
©Gertrude Battaly, 2014
Class Notes
σ's
Homework
11.2 Inferences for σ's, 2 Populations
geogebra
More analytical approach
Use the F­ Distribution with Hypothesis Test F ­ Distribution
1. Ratio of variations
∎ ANOVA F = MSTR MSE
∎ Two σ's Test F = s12 s22 2. Total area under curve = 1
3. Starts at 0.
4. Right­skewed.
F­Distribution Table
∎ 8 pages long
∎ df numerator across top
∎ df denominator on sides, with
α = 0.10, 0.05, 0.025, 0.01, 0.005 Class Notes: Prof. G. Battaly, Westchester Community College, NY
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© G. Battaly 2014
©Gertrude Battaly, 2014
Class Notes
σ's
Homework
2
Comparing Standard Deviations from Two Populations
May 05, 2014
11.2 Inferences for σ's, 2 Populations
What F ­ value can we expect? 1. s1 is best estimate of σ1, s2 is best estimate of σ2
2. If σ1 = σ2 then F = s12 should be close to 1 s22
If σ1 < σ2 then F = s12 should be < 1 s22
If σ1 > σ2 then F = s12 should be >1 s22
3. Since we expect variation in sample stdev's, we do NOT expect that the sample F value =1 exactly, not even if σ1 = σ2
4. Use hypothesis test to decide how much less than or greater than 1 F needs to be to decide between null and alternative hypotheses.
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
Class Notes
©Gertrude Battaly, 2014
Homework
11.2 Inferences for σ's, 2 Populations
Two Standard Deviation F­Test
Step 1: Assumptions: 1. SRS, 2. Indep samples 3. ND or large samples H0: σ1 = σ2
Ha: σ1 < σ2
or
σ1 ≠ σ2
σ1 > σ2
or
Step 2: Decide α
Step 3: Compute F = s12 s22 Step 4: Find CV(s) OR Find p­value: left­tailed p = Fcdf (0, Ftest, dfn, dfd)
2­tailed p = 2*Fcdf (0, Ftest, dfn, dfd) if F <1 or p = 2(1 ­ Fcdf (0, Ftest, dfn, dfd) ) if F >1
right­tailed p = 1­ Fcdf (0, Ftest, dfTR, dfE)
Step 5: Decide whether to reject H
0 or not
Step 6: Verbal interpretation
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
© G. Battaly 2014
©Gertrude Battaly, 2014
Class Notes
Homework
3
Comparing Standard Deviations from Two Populations
May 05, 2014
11.2 Inferences for σ's, 2 Populations
An independent s.r.s of infants was taken. 10 infants were treated for pulmonary hypertension (PH), 25 infants were not treated (control). Head circumferences were measured:
Based on this data, at the 5% significance level, does a difference in variation exist between infants treated for PH and those not treated? (Note: A normal probability plot is approximately linear. s1=1.907, s2=1.594)
Two Standard Deviation F­Test
Step 1: Assumptions: 1. SRS, 2. Indep samples 3. ND or large samples H0: σ1 = σ2
Ha: σ1 < σ2
or
σ1 ≠ σ2
or
σ1 > σ2
Step 2: Decide α
Step 3: Compute F = s12 s22 Step 4: Find CV(s) OR Find p­value: left­tailed p = Fcdf (0, Ftest, dfn, dfd)
2­tailed p = 2*Fcdf (0, Ftest, dfn, dfd) if F <1 or p = 2(1 ­ Fcdf (0, Ftest, dfn, dfd) ) if F >1
right­tailed p = 1­ Fcdf (0, Ftest, dfTR, dfE)
Step 5: Decide whether to reject H
0 or not
Step 6: Verbal interpretation
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
©Gertrude Battaly, 2014
Homework
Class Notes
Two Standard Deviation F­Test
11.2 Inferences for σ's, 2 Populations
An independent s.r.s of infants was taken. 10 infants were treated for pulmonary hypertension (PH), 25 infants were not treated (control). Head circumferences were measured:
Based on this data, at the 5% significance level, does a difference in variation exist between infants treated for PH and those not treated? (Note: A normal probability plot is approximately linear. s1=1.907, s2=1.594)
Step 1: Assumptions: 1. SRS, 2. Indep samples 3. ND or large samples H0: σ1 = σ2
Ha: σ1 < σ2
or
σ1 ≠ σ2
or
σ1 > σ2
Step 2: Decide α
Step 3: Compute F = s12 s22 Step 4: Find CV(s) OR Find p­value: left­tailed p = Fcdf (0, Ftest, dfn, dfd)
2­tailed p = 2*Fcdf (0, Ftest, dfn, dfd) right­tailed p = 1­ Fcdf (0, Ftest, dfTR, dfE)
Step 5: Decide whether to reject H
0 or not
Step 1: Assumptions: 1. SRS, 2. Indep samples 3. ND or large samples Step 6: Verbal interpretation
H0: σ1 = σ2 H : σ ≠ σ
a
1
2
Step 2: α = 0.05
Step 3: Compute F = s12 = 1.907 = 1.431
s22 1.594
Step 4: p = 0.4601 left­tailed p = Fcdf (0, Ftest, dfn, dfd)
2­tailed p = 2*Fcdf (0, Ftest, dfn, dfd) if F <1 or p = 2(1 ­ Fcdf (0, Ftest, dfn, dfd) ) if F >1
right­tailed p = 1­ Fcdf (0, Ftest, dfTR, dfE) Step 5: p = 0.4601 > 0.05 = α ∴ do NOT Reject H0
Step 6: Based on this data, there is no difference in variation among head circumferences of infants treated for PH and those not treated.
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
© G. Battaly 2014
©Gertrude Battaly, 2014
Class Notes
Homework
4
Comparing Standard Deviations from Two Populations
May 05, 2014
11.2 Inferences for σ's, 2 Populations
Soil scientists have measured the arsenic concentration in the soil using two different methods. Ten independent simple random samples were taken using each of the two methods. The scientists want to determine which methods results in more precise data. The more precise method would have a lower standard deviation since it would result in more consistent outcomes when measuring mean amounts of arsenic. Data for the two methods includes:
Method Mean (ppm) s (ppm) n
1 6.7 0.8 10
2 8.2 1.2 10
At a 5% significance level, is Method 1 more precise than Method 2? (A normal prob plot appears approx linear.)
Two Standard Deviation F­Test
Step 1: Assumptions: 1. SRS, 2. Indep samples 3. ND or large samples H0: σ1 = σ2
Ha: σ1 < σ2
or
σ1 ≠ σ2
or
σ1 > σ2
Step 2: Decide α
Step 3: Compute F = s12 s22 Step 4: Find CV(s) OR Find p­value: left­tailed p = Fcdf (0, Ftest, dfn, dfd)
2­tailed p = 2*Fcdf (0, Ftest, dfn, dfd) right­tailed p = 1­ Fcdf (0, Ftest, dfTR, dfE)
Step 5: Decide whether to reject H
0 or not
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
©Gertrude Battaly, 2014
Homework
Class Notes
11.2 Inferences for σ's, 2 Populations
Step 1: Assumptions: 1. SRS, 2. Indep samples 3. ND H0: σ1 = σ2
Ha: σ1 < σ2
Step 6: Verbal interpretation
Soil scientists have measured the arsenic concentration in the soil using two different methods. Ten independent simple random samples were taken using each of the two methods. The scientists want to determine which methods results in more precise data. The more precise method would have a lower standard deviation since it would result in more consistent outcomes when measuring mean amounts of arsenic. Data for the two methods includes:
Step 2: α = 0.05
Step 3: Compute F = s12 = 0.8 = 0.444
Method Mean (ppm) s (ppm) n
1 6.7 0.8 10
2 8.2 1.2 10
At a 5% significance level, is Method 1 more precise than Method 2? (A normal prob plot appears approx linear.)
s22 1.2
Step 4: p = 0.121 left­tailed p = Fcdf (0, Ftest, dfn, dfd)
Step 5: p = 0.121 > 0.05 = α ∴ do NOT Reject H0
Step 6: Based on this data, conclude that there is no difference in
variation among the two groups. Therefore, there is no difference in precision between the two methods.
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Statistics Home Page
© G. Battaly 2014
©Gertrude Battaly, 2014
Class Notes
Homework
5
Attachments
Statistical Tables.pdf