Question 1 (a). Find the probabilities listed here. In each instance, show detailed calculations, sketch the Normal curve, and shade the “area under the curve” associated with the probability.
i. Pr(Z > −0.64) ii. Pr(−0.64 < Z < 1.65). iii. between 1.42 and 1.25 (b). The SAT and ACT are standardized tests for college admission in the United States. Both tests include components that measure reading comprehension. Suppose that SAT critical reading scores are Normally distributed with a mean of 510 and standard deviation of 115. In contrast, ACT reading scores are Normally distributed with a mean of 20.5 and standard deviation of 5. Sam takes the SAT reading test and scores 660. Dave takes the ACT test and scores 28. Who had the superior score, Sam or Dave? Use appropriate formula, sketch the curves, and provide detailed calculations showing how the formula leads to your answers. Question 2 In a public hospital having large number of patients with psychosocial problems, 36 in-patients were selected through simple random sampling and were asked to complete a depression scale. Scores derived from the scale are known to be Normally distributed with μ = 1.3 and σ = 0.3, with high scores indicating worsening cases. a) Test a two-sided alternative hypothesis, given that the mean score from 25 patients was computed to be 1.4. (Hint: Arrange your solution thus: A. Hypothesis; B. Choice of Test statistic; C. Evaluation of Test Statistic; D. P-value; E. Significance (or decision); F. Conclusion) b) Calculate the Z-statistic for the one-sided alternative hypothesis for the test in question 1a?. c) Convert the Zstat in question 1b to a P-value, and interpret the results.