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Outline. Single sample test of hypothesis about a population variance The  2 test rejection region Example. Test of hypothesis about a population variance. This is a test that answers a question about a single population.

byron-stewart
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Outline

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  1. Outline • Single sample test of hypothesis about a population variance • The 2 test rejection region • Example 2

  2. Test of hypothesis about a population variance • This is a test that answers a question about a single population. • The question is not about the population mean, µ – it is about the variance, 2. • Usually, the question will be whether the population variance has changed from some historical value. 2

  3. Test of hypothesis about a population variance • The ratio • (n-1) s2 • 2 • has a sampling distribution called 2 (“chi-square”) when the population sampled is normal. 2

  4. 2 distributions for 1, 2, 3, 5, and 10 degrees of freedom – notice that the shape of the distribution depends upon d.f. As d.f. goes up, the sampling distribution becomes increasingly normal. df = 10 2

  5. The tabled values of 2 locate an area  in the upper tail of the distribution. 2

  6. 2Hypothesis Test • HO: 2 = 2 HO: 2 = 2 • HA: 2 < 2 HA: 2≠ 2 • (or 2 > 2) • Test statistic: 2 = (n-1)s2 • 2 2

  7. 2 Hypothesis Test – Rejection Regions • HO: 2 = 2 HO: 2 = 2 • HA: 2 < 2 HA: 2≠ 2 • (or 2 > 2) • 2 <21- 2 < 21-/2 • or 2 >2 2 > 2/2 • Where 2 is based on (n-1) degrees of freedom. 2

  8. Example 2 • Tetris is a computer game requiring some spatial information-processing skills and good eye-hand coordination, either or both of which may improve with practice. Six people who had never previously played Tetris were tested on the game at the beginning (Test 1) and at the end (Test 2) of a 2-week period during which they played Tetris for one hour each day. Their Tetris scores on the two testing sessions appear on the next slide. Paired Differences

  9. Example 2 • a. Did the subjects’ Tetris scores improve significantly from Test 1 to Test 2 (α = .05)? • b. Is the variance of the subjects’ Test 2 scores significantly different from 400,000, the variance among the population of experts at Tetris (α = .05)? Paired Differences

  10. Example 2a • Subject Test 1 Test 2 Diff D2 • 1 3025 5642 2617 6848689 • 2 4120 5117 997 994009 • 3 2675 4333 1658 2748964 • 4 6715 6026 -689 474721 • 5 1997 5429 3432 11778624 • 6 4807 4807 0 Paired Differences

  11. Example 2b • HO: σ2 = 400,000 • HA: σ2 ≠ 400,000 • Test statistic: Χ2 = (n-1)s2 • σ2 • Rejection region: Χ2 > Χ2(.025) = 12.8325 or • Χ2 < Χ2(.975) = .831211 Paired Differences

  12. Example 2b • Χ2 = 5 (367831.068) = 4.598 • 400,000 • Decision: do not reject HO. No evidence that the Test 2 variance differs from 400,000. • Note: 367831.068 = s2 for Test 2 scores, NOT for the difference scores. Paired Differences

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