is the standard deviation of the differences between the paired or related

# Is the standard deviation of the differences between

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𝑑 is the standard deviation of the differences between the paired or related observations 𝑛 is the number of paired observations Two Sample Tests: Dependent Samples The standard deviation of the differences will be computed with ? 𝑑 = Σ 𝑑 − ҧ 𝑑 2 𝑛 − 1 Example: Dependent Samples Nickel Savings and Loan wishes to compare the two companies it uses to appraise the value of residential homes. Nickel Savings selected a sample of 10 residential properties and scheduled both firms for an appraisal. The results, reported in \$,000 are At the .05 significance level, can we conclude that there is a difference between the firms’ mean appraised home values? Example: Dependent Samples Step : State ? 0 and ? 1 Step : Select level of significance 𝛼 = 0.05 (given) Step : Select test statistics ? distribution since 𝜎 is unknown ? 0 : 𝜇 𝑑 = 0 ? 1 : 𝜇 𝑑 ≠ 0 Two-tailed test Example: Dependent Samples Step : Formulate decision rule Two-tailed, 𝛼 = 0.05 , 𝑑𝑓 = 𝑛 − 1 = 10 − 1 = 9 Reject ? 0 if computed t > 2.262 if computed t < −2.262 ? = 2.262 Example: Dependent Samples Step : Make a decision ҧ 𝑑 = 46 10 = 4.6 ? 𝑑 = 174.4 10 − 1 = 4.402 ? = ҧ 𝑑 ? 𝑑 / 10 = 4.6 4.402/ 10 = 3.305 As computed ? > critical ? , reject ? 0 Reject ? 0 if t > 2.262 if t < −2.262 Example: Dependent Samples Step : Interpret result There is a difference between the firms’ mean appraised home values. The largest difference is \$12,000 for Home 3. What is the 𝑝 -value? We need the probability value of ? to be closest to 3.305, with the 𝑑𝑓 value of 9. This value is between 3.250 and 4.781, thus corresponds to the significance level of .010. ∴ 𝑝 -value < 0.01 Therefore, there is a small likelihood that the ? 0 is true. 