1 Objective Compare of two population variances using two samples from each population. Hypothesis Tests and Confidence Intervals of two variances use the F-distribution Section 9.5 Comparing Variation in Two Samples

2 (1) The two populations are independent (2) The two samples are random samples (3) The two populations are normally distributed (Very strict!) Requirements All requirements must be satisfied to make a Hypothesis Test or to find a Confidence Interval

3 Important The first sample must have a larger sample standard deviation s 1 than the second sample. i.e. we must have s 1 ≥ s 2 If this is not so, i.e. if s 1 < s 2, then we will need to switch the indices 1 and 2

4 Notation σ 1 First population standard deviation s 1 First sample standard deviation n 1 First sample size σ 2 Second population standard deviation s 2 Second sample standard deviation n 2 Second sample size Note: Use index 1 on sample/population with the larger sample standard deviation (s)

5 Tests for Two Proportions H 0 : σ 1 = σ 2 H 1 : σ 1 ≠ σ 2 Two tailedRight tailed Note: We do not consider σ 1 < σ 2 (since we used indexes 1 and 2 such that s 1 is larger) Note: We only test the relation between σ 1 and σ 2 (not the actual numerical values) The goal is to compare the two population variances (or standard deviations) H 0 : σ 1 = σ 2 H 1 : σ 1 > σ 2

6 The F-Distribution Similar to the χ 2 -dist. Not symmetric Non-negative values (F ≥ 0) Depends on two degrees of freedom df 1 = n 1 – 1 (Numerator df ) df 2 = n 2 – 1 (Denominator df )

7 The F-Distribution On StatCrunch: Stat – Calculators – F df 1 = n 1 – 1 df 2 = n 2 – 1

8 s1 s1 F = s2 s2 2 2 Test Statistic for Hypothesis Tests with Two Variances Where s 1 2 is the first (larger) of the two sample variances Because of this, we will always have F ≥ 1

9 If the two populations have equal variances, then F = s 1 2 /s 2 2 will be close to 1 (Since s 1 2 and s 2 2 will be close in value) If the two populations have different variances, then F = s 1 2 /s 2 2 will be greater than 1 (Since s 1 2 will be larger than s 2 2 ) Use of the F Distribution

10 Conclusions from the F-Distribution Values of F close to 1 are evidence in favor of the claim that the two variances are equal. Large values of F, are evidence against this claim (i.e. it suggest there is some difference between the two)

11 Steps for Performing a Hypothesis Test on Two Independent Means Write what we know Index the variables such that s 1 ≥ s 2 (important!) State H 0 and H 1 Draw a diagram Find the Test Statistic Find the two degrees of freedom Find the Critical Value(s) State the Initial Conclusion and Final Conclusion

12 Example 1 Below are sample weights (in g) of quarters made before 1964 and weights of quarters made after When designing coin vending machines, we must consider the standard deviations of pre-1964 quarters and post-1964 quarters. Use a 0.05 significance level to test the claim that the weights of pre-1964 quarters and the weights of post-1964 quarters are from populations with the same standard deviation. Claim: σ 1 = σ 2 using α = 0.05

13 H 0 : σ 1 = σ 2 H 1 : σ 1 ≠ σ 2 Test Statistic Critical Value Initial Conclusion: Since F is in the critical region, reject H 0 Final Conclusion: We reject the claim that the weights of the pre-1964 and post-1964 quarters have the same standard deviation Example 1 Using StatCrunch: Stat – Calculators – F F α/2 = F = Two-Tailed H 0 = Claim n 1 = 40 n 2 = 40 α = 0.05 s 1 = s 2 = (Note: s 1 ≥s 2 ) 2 2 Degrees of Freedom df 1 = n 1 – 1 = 39 df 2 = n 2 – 1 = 39 F α/2 = F = F is in the critical region

14 H 0 : σ 1 = σ 2 H 1 : σ 1 ≠ σ 2 Initial Conclusion: Since P-value is less than α (0.05), reject H 0 Final Conclusion: We reject the claim that the weights of the pre-1964 and post-1964 quarters have the same standard deviation Example 1 Two-Tailed H 0 = Claim n 1 = 40 n 2 = 40 α = 0.05 s 1 = s 2 = (Note: s 1 ≥s 2 ) Null: variance ratio= Alternative Sample 1: Variance: Size: Sample 2: Variance: Size: ● Hypothesis Test P-value = Stat → Variance → Two sample → With summary s 1 2 = s 2 2 = ≠1≠