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Section 9.4 Inferences About Two Means (Matched Pairs)

Section 9.4 Inferences About Two Means (Matched Pairs). Objective Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means use the t -distribution. Definition.

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Section 9.4 Inferences About Two Means (Matched Pairs)

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  1. Section 9.4Inferences About Two Means(Matched Pairs) Objective Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means use the t-distribution

  2. Definition Two samples are dependentif there is some relationship between the two samples so that each value in one sample is paired with a corresponding value in the other sample. Two samples can be treated as the matched pairs of values.

  3. Examples • Blood pressure of patients before they are given medicine and after they take it. • Predicted temperature (by Weather Forecast) and the actual temperature. • Heights of selected people in the morning and their heights by night time. • Test scores of selected students in Calculus-I and their scores in Calculus-II.

  4. Example 1 First sample: weights of 5 students in April Second sample: their weights in September These weights make 5 matched pairs Third line: differences between April weights and September weights (net change in weight for each student, separately) In our calculations we only use differences (d), not the values in the two samples.

  5. dIndividual difference between two matched paired values μdPopulation mean for the difference of the two values. nNumber of paired values in sample dMean value of the differences in sample sdStandard deviation of differences in sample Notation

  6. (1) The sample data are dependent (i.e. they make matched pairs) (2) Either or both the following holds: The number of matched pairs is large (n>30) or The differences have a normal distribution Requirements All requirements must be satisfied to make a Hypothesis Test or to find a Confidence Interval

  7. Tests for Two Dependent Means Goal: Compare the mean of the differences H0:μd=0 H1:μd≠0 H0:μd=0 H1:μd<0 H0:μd=0 H1:μd>0 Two tailed Left tailed Right tailed

  8. d –µd t= sd n degrees of freedom: df = n – 1 Finding the Test Statistic Note: md= 0according to H0

  9. Test Statistic Degrees of freedomdf= n – 1 Note: Hypothesis Tests are done in same way as in Ch.8-5

  10. Steps for Performing a Hypothesis Test on Two Independent Means • Write what we know • State H0 and H1 • Draw a diagram • Calculate the Sample Stats • Find the Test Statistic • Find the Critical Value(s) • State the Initial Conclusion and Final Conclusion Note: Same process as in Chapter 8

  11. Example 1 Assume the differences in weight form a normal distribution. Use a 0.05 significance level to test the claim that for the population of students, the meanchangein weight from September to April is 0 kg (i.e. on average, there is no change) Claim: μd = 0 using α = 0.05

  12. d Data: -1 -1 4 -2 1 Example 1 H0:µd=0 H1:µd≠0 t-dist. df = 4 Two-Tailed H0 =Claim t = 0.186 -tα/2 = -2.78 tα/2 = 2.78 Sample Stats n = 5 d = 0.2sd = 2.387 Use StatCrunch: Stat – Summary Stats – Columns Test Statistic Critical Value tα/2 = t0.025 = 2.78 (Using StatCrunch, df = 4) Initial Conclusion:Since t is not in the critical region, accept H0 Final Conclusion: We accept the claim that mean change in weight from September to April is 0 kg.

  13. d Data: -1 -1 4 -2 1 Example 1 Sample Stats H0:µd=0 H1:µd≠0 n = 5 d = 0.2sd = 2.387 Use StatCrunch: Stat – Summary Stats – Columns Two-Tailed H0 =Claim Stat → T statistics→ One sample → With summary ● Hypothesis Test Sample mean: Sample std. dev.: Sample size: 0.2 2.387 5 Null: proportion= Alternative 0 ≠ P-value = 0.8605 Initial Conclusion:Since P-value is greater than α (0.05), accept H0 Final Conclusion: We accept the claim that mean change in weight from September to April is 0 kg.

  14. Confidence Interval Estimate We can observe how the two proportions relate by looking at the Confidence Interval Estimate of μ1–μ2 CI = (d – E,d + E)

  15. Find the 95% Confidence Interval Estimate of μd from the data in Example 1 Example 2 Sample Stats n = 5 d = 0.2 sd = 2.387 tα/2 = t0.025 = 2.78 (Using StatCrunch, df = 4) CI =(-2.8, 3.2)

  16. Find the 95% Confidence Interval Estimate of μd from the data in Example 1 Example 2 Sample Stats n = 5 d = 0.2 sd = 2.387 Stat → T statistics→ One sample → With summary ● Confidence Interval Sample mean: Sample std. dev.: Sample size: 0.2 2.387 5 Level: 0.95 CI =(-2.8, 3.2)

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