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Lesson #20 Tests for a Single Mean

Lesson #20 Tests for a Single Mean. under H 0. under H 0. Test of H 0 : m = m 0 , when s 2 is known. (one-sample Z-test for the mean). under H 0. Test of H 0 : m = m 0 , when s 2 is known. (one-sample Z-test for the mean). under H 0. test statistic. Distribution of

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Lesson #20 Tests for a Single Mean

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  1. Lesson #20 Tests for a Single Mean

  2. under H0 under H0 Test of H0: m = m0, when s2 is known (one-sample Z-test for the mean)

  3. under H0 Test of H0: m = m0, when s2 is known (one-sample Z-test for the mean) under H0 test statistic

  4. Distribution of Z0 under H1 Distribution of Z0 under H0 , N(0,1) a Critical value H1: m > m0 0 Reject if Z0 is relatively large Rejection region = Z1-a Reject H0 if Z0 > Z1-a P(Z0 > critical value | H0 true) = P(reject H0 | H0 true) = a

  5. Distribution of Z0 under H1 Distribution of Z0 under H0 , N(0,1) a b H1: m > m0 0 Rejection region Z1-a P(Z0 < critical value | H1 true) = P(fail to reject H0 | H1 true) = b

  6. Distribution of Z0 under H1 Distribution of Z0 under H0 , N(0,1) Power b H1: m > m0 0 Rejection region Z1-a Power = P(reject H0 | H1 true) = 1 - b

  7. Distribution of Z0 under H1 Distribution of Z0 under H0 , N(0,1) a p-value Observed Z0 H1: m > m0 0 p-value is the area under N(0,1) curve above observed Z0 Rejection region Z1-a  reject H0  p-value = P(Z > Z0) Reject H0 if p-value < a

  8. H0: m = m0 vs. H1: m > m0 Reject H0 if Z0 > Z1-a p-value = P(Z > Z0) H0: m = m0 vs. H1: m < m0 Reject H0 if Z0 < Za = -Z1-a p-value = P(Z < Z0)

  9. In all cases, reject H0 if p-value < a H0: m = m0 vs. H1: mm0 Z0 > Z1-a/2 Reject H0 if or Z0 < Za/2 = -Z1-a/2 Reject H0 if |Z0| > Z1-a/2 p-value = 2P( Z > |Z0| )

  10. Hypothesis Testing Steps 1) Determine hypotheses 2) Decide on a ( .01 , .05 , .10 ) 3 & 4) State rejection region, calculate test statistic (or) Calculate test statistic and p-value 5) Make decision (reject or not reject) 6) Write conclusions (interpret results), in the context of the problem

  11. under H0 under H0 Test of H0: m = m0, when s2 is unknown (one-sample t-test for the mean) If the variance is known, the test statistic was If the variance is unknown, the test statistic is

  12. H0: m = m0 vs. H1: m > m0 Reject H0 if t0 > t(n-1),1-a p-value = P(t(n-1) > t0) H0: m = m0 vs. H1: m < m0 Reject H0 if t0 < -t(n-1),1-a p-value = P(t(n-1) < t0) H0: m = m0 vs. H1: mm0 Reject H0 if |t0| > t(n-1),1- a/2 p-value = 2P(t(n-1) > |t0| )

  13. 1) 2) 3) 4) 5) 6) H0: m = 7250 H1: m < 7250 a = .05 Reject H0 if t0 < -t(14),.95 = -1.761 4767 - 7250 = -3.00 3204 Reject H0 Conclude mean white blood cell count among Humans infected with E. canis is less than 7250, the mean of uninfected humans.

  14. t0 = H0: m = 100 H1: m 100 a = .05 Reject H0 if |t0| > t(17),.975 = 2.110 112.778 - 100 = 3.76 14.424 Reject H0 Conclude mean percent ideal body weight among insulin-dependent diabetics is greater than 100.

  15. actual p-value = (SAS two-sided p-value) actual p-value = 1 - (SAS two-sided p-value) SAS yields p-values for two-sided tests when performing t-tests. If you are doing a one-sided test, you have to adjust the SAS results to get the actual p-value. If the results are in the direction of H1 If the results are in the opposite direction of H1

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