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Chapter 11

Chapter 11. Hypothesis Test about Population Variance ( Standard Deviation ). Variance. Variance Squared standard deviation Sample variance: s 2 Population variance: σ 2 Measuring the spread (the variability , the dispersion ) of the data around the mean

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Chapter 11

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  1. Chapter 11 Hypothesis Test about Population Variance (Standard Deviation)

  2. Variance • Variance • Squared standard deviation • Sample variance: s2 • Population variance: σ2 • Measuring the spread (the variability, the dispersion) of the data around the mean • A low variance indicates that the data is clustered around the mean (less variation, more consistency) • A high variance indicates that the data are widely spread out of the mean (more variation, less consistency)

  3. Variance • Firm A: Mean= 5%, Standard Deviation (SD)= 3.3%, Variance = (3.3%)2 • Firm B: Mean= 5%, Standard Deviation (SD)= 1.0%, Variance = (1.0%)2 0 0

  4. Hypothesis Tests for Population Variance • Two hypothesis test procedures for population variance: • Hypothesis test for a single population variance • Hypothesis test for two population variances

  5. Hypothesis test for a single population variance • Example: • A lumber company has claimed that the standard deviation for the lengths of their 6 foot boards is 0.5 inches or less. To test their claim, a random sample of 17 six-foot boards is selected; and it is determined that the standard deviation of the sample is 0.4. Do the results of the sample support the company's claim? • In this example, the lumber company claims that the lengths of their 6-foot boards are quite consistent (not much variation in the lengths of their products) because the standard deviation is small (≤ 0.5)

  6. Hypothesis test for a single population variance • Ho: σ2 = value(σ02) • Ha: σ2 ≠ value (σ02) • Ho: σ2 ≥ value (σ02) • Ha: σ2 < value (σ02) • Ho: σ2 ≤ value (σ02) • Ha: σ2 > value (σ02)

  7. Hypothesis test for a single population variance • Step 1.Formulate H0 and Ha in terms of the population variance (NOT standard deviation) • Step 2. Compute the sample variance s2 (formula 11.1 on page 450) Assume that s2 is an estimate for σ2 . If the sample variance s2 is consistent with Ha, then Ha is supported. So next, compute the probability that this result is wrong.

  8. Hypothesis test for a single population variance • Step 3. Compute the chi-square (χ2) by using Formula 11.8 on page 454 • σ02is the number in the hypothesis. DO NOT SQUAREthis number • Step 4. Compute the p-value by using (1) the χ2, the d.f. (d.f. = n – 1) , and the p-value calculator from χ2 (http://www.danielsoper.com/statcalc/calc11.aspx) • If Ha has >, use upper-tail p-value • If Ha has <,use lower-tail p-value • If Ha has ≠, use two-tailp-value

  9. Hypothesis test for a single population variance • Step 5. Determineα • Step 6. If p-value ≤ α, accept the result supporting Ha to be true

  10. Step 4. Compute the p-value • The calculator only gives you the upper-tail p-value • To compute the lower-tail p-value: • lower-tail p-value = 1 – (upper-tail p-value) • To compute the two-tail p-value: • Step 1. Compute the upper-tail p-value and the lower-tail p-value • Step 2. Select the smaller value between those two p-values • Step 3. Doublethat value for the two-tail p-value

  11. Hypothesis test for a single population variance • Examples: • Problem 10 on page 459 • Variance is squared standard deviation • There is no hypothesis test for standard deviation • To test a statement/claim about standard deviation, you have to transform the standard deviation into variance and do the hypothesis test for variance • n = 36 ; s2 = (0.222)2 ; α= 0.05

  12. How to formulate the hypotheses? • Step 1. Identify what the σ2 (population variance) represents • Step 2. Find the sentence in the problem that says that the σ2 (population variance) is: • Greater than (exceed, has increased), • Less than (fewer than, has been reduced), • Not equal to (is not, different from, has changed) • Step 3. Translate the sentence into statistical/mathematical statement: • σ2 > value, • σ2 < value, • σ2 ≠ value

  13. Hypothesis test for two population variance • This testing procedure is used to test: • If population one has greatervariance than population two • If population one has differentvariance from the population two

  14. Hypothesis test for two population variance • Example: • The standard deviation of the ages of a sample of 16 executives from the northern states was 8.2 years; while the standard deviation of the ages of a sample of 25 executives from the southern states was 12.8 years. • Test to see if executives from the northern states are less diverse than the ones from the southern states.

  15. Hypothesis test for two population variance • Two-Tail Test • Ho: σ12= σ22 • Ha: σ12≠ σ22 Upper-Tail Test • Ho: σ12 ≤ σ22 • Ha: σ12 > σ22 Attention: • Always use the larger sample variance (s2) as σ12 • σ12 is the population variance that represents the larger sample variance (s2) • Never use lower-tail test (never use Ha: σ12 < σ22)

  16. ! For TWO population variance, NEVER use Ha: σ12 < σ22

  17. Hypothesis test for two population variance • Step 1.Formulate H0 and Ha in terms of the two population variances (Remember: Ha: σ12 > σ22 ORHa: σ12 ≠ σ22 , NEVER Ha: σ12 < σ22) • Step 2. Compute the sample variance s2 (formula 11.1 on page 450) Assume that s2 is an estimate for σ2 . If the sample variance s2 is consistent with Ha, then Ha is supported. So next, compute the probability that this result is wrong.

  18. Hypothesis test for two population variance • Step 3. Compute Fby using Formula 11.10 on page 461 • Step 4. Compute the p-value by using (1) the F, the d.f. of the two populations(d.f. = n – 1; numerator d.f. is d.f. for the larger variance; denominator d.f. is the d.f. for the smaller variance) , and the p-value calculator from F(Use the P from F calculator in http://www.graphpad.com/quickcalcs/pvalue1.cfm) • If Ha has > use Upper-tail p-value • If Ha has ≠ use Two-tail p-value

  19. Hypothesis test for two population variance • Step 5. Determine α • Step 6. If p-value ≤ α, accept the result supporting Ha to be true

  20. Step 3. Compute F by using Formula 11.10 on page 461 • Remember: • Which variance is the s12? • The larger/greatersamplevariance • Which variance is the s22? • The smaller sample variance • HENCE, F is always > 1. If F is < 1, you have to switch the two variances

  21. Step 4. Compute the p-value • Which d.f. is the d.f.1(the numerator d.f.)? • The d.f. computed from the sample that has the larger/greater variance • d.f.1 is NOT always the larger d.f. • Which d.f. is the d.f.2(the denominator d.f.)? • The d.f. computed from the sample that has the smaller variance • d.f.2 is NOT always the smaller d.f.

  22. Step 4. Compute the p-value • The calculator only gives you the upper-tail p-value • To compute the two-tail p-value: • Step 1. Compute the upper-tail p-value and the (1 – the upper-tail p-value ) • Step 2. Select the smaller value between those two p-values • Step 3.Double that value for the two-tail p-value

  23. Hypothesis test for two population variance • Examples: • Problem 16 • Which variance is the s12? • Which variance is the s22? • Which d.f. is the d.f.1 (the numerator d.f.)? • Which d.f. is the d.f.2 (the denominator d.f.)? • Problem 18 • Which variance is the s12? • Which variance is the s22? • Which d.f. is the d.f.1 (the numerator d.f.)? • Which d.f. is the d.f.2 (the denominator d.f.)?

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