Significance tests #2 - Questions and Answers

Question 1 The average life span of a Golden Retriever is said to be 9.6 years. In a random sample of 15 Golden Retrievers, the following life spans were found: 6.8 10.2 9.5 13 9.9 7.0 8.3 8.9 11.2 10 9.8 13 9.8 9.5 8.7 At a 5% significance level, if we do a complete hypothesis test to test eh claim that the average life span of Golden Retrievers is 9.5, the p-value and conclusion are: p-value = 0.8187, fail to reject the null hypothesis p-value = 0.8153, fail to reject the null hypothesis p-value = 0.8187, reject the null hypothesis p-value = 0.8153, reject the null hypothesis Answer 1 Given: The average lifespan of a golden retriever is a set to be 9.6 years in the random sample of 15 golden retriever the following life span were found. From tbe given data Sample size, n = 15 Σx = 145.6 Sample mean, x̄ = Σx / n = 145.6 / 15 = 9.7067 Sample standard deviation, s = 1.7689 Population mean, μ = 9.6 Hypothesis test: The null and alternative hypothesis is H₀ : μ = 9.6 Hₐ : μ ≠ 9.6 Test statistics is z = (x̄ - μ) / (s/√n) = (9.7067 - 9.6) / (1.7689/√15) = 0.2336 ∴ z = 0.23 p-value for two tailed: p-value = 2P(z > 0.23) = 2 × 0.409046...(from z-table) = 0.8181 ≈ 0.8187 ∴ p-value = 0.8187 Answer 2 Since p-value is greater than significance level 0.05, we fail to reject null hypothesis. p-value = 0.8187, fail to reject the null hypothesis. Therefore the correct option is A Question 2 When blocking is used, discuss how the blocks are likely to be selected and whether their selection method has an impact on the analysis of variance procedure. Answer Blocks are groups of experimental units. Each block is homogeneous within. Typically in design of experiments blocks are selected on the basis of a specific characteristic like gender, age group etc. A blocking factor is a source of variability that is not of primary interest to the experimenter. Blocking reduces unexplained variability. Its principle lies in the fact that variability which cannot be overcome is confounded with a higher order interaction (which is of least importance) to eliminate its influence on the end product. When we can control nuisance factors (confounders), blocking can be used to reduce or eliminate the contribution to experimental error contributed by nuisance factors. The basic concept is to create homogeneous blocks in which the nuisance factors are held constant and the factor of interest is allowed to vary. Within blocks, it is possible to assess the effect of different levels of the factor of interest without having to worry about variations due to changes of the block factors, which are accounted for in the analysis. Question 3 (a) Find the F-statistic from ANOVA table. Find the p-value. (b) Explain about three p-values for the three tests in parts(a), (b),and(c). Answer (a)From the table of ANOVA, the F-statistic for the test is 6.50, and the p-value is 0.015. (b)From the computer output, the p-value of 0.015 is equal for all three tests. the regression model with single predictor, the p-value is same for all three tests. Question 4 Critical Thinking: One-Tailed versus Two-Tailed Tests For the same data and null hypothesis, is the P-value of a one-tailed test (right or left) larger or smaller than that of a two-tailed test? Explain. Answer Calculation: A one-tailed test has a lower P-value than a two-tailed test. Because the two-tailed test includes the area in both tails and the P-value is twice that of the one-tailed (left or right) test. For the same data, P-value for one tailed test is 0.0127, then the P-value for two tailed test will be 0.0254. Question 5 Describe the five-step p-value approach to hypothesis testing. Answer The p-value approach tests the hypothesis based on the p-value . The five-step critical value approach to hypothesis testing is: Step 1: Formulate the null hypothesis H₀ and alternative hypothesis H₁. Step 2: Select the level of significance α and the sample size n. Step 3: Select the appropriate test statistic and the distribution of the sample. Step 5: Gather and organize the sample data and calculate the value of the test statistic and p-value. Step 6: Make statistical decision. If the p-value is greater than the level of significance (α), then the null hypothesis is not rejected. If the p-value is less than the level of significance (α), then the null hypothesis is rejected.