Example A: The SAT is designed so that the observed scores will have normal distribution. Do the 162 scores listed below support this hypothesis? (α = 0.05) 800 800 790 780 760 760 760 760 750 740 740 740 740 740 740 730 720 720 720 720 720 710 710 710 700 700 700 700 700 700 700 700 700 700 700 690 690 690 690 690 690 690 690 680 680 680 680 680 680 680 680 670 670 670 670 660 660 660 650 650 650 650 650 650 650 650 650 650 640 640 640 640 640 640 640 640 640 640 630 630 630 630 630 630 630 630 620 620 620 620 610 610 610 610 600 600 600 600 590 590 590 580 580 580 570 570 570 570 570 570 570 570 560 560 560 560 560 560 550 550 550 540 540 540 540 540 540 540 530 530 530 530 530 530 520 520 520 520 510 510 500 500 490 490 480 470 460 450 440 440 430 430 420 410 410 410 410 400 400 390 390 360 hypotheses: Prior to sampling, it is presumed that the population mean μ is 600 with standard deviation of 100. As noted at the end of Lecture 14.1b, the eight z-intervals (-0, -1.15), [-1.15, -0.675), [-0.675, -0.32), [-0.32,0), [0, 0.32), [0.32, 0.675), [0.675, 1.15) and [1.15, ∞) have a probability of 1/8 each, i.e. the probability is uniform for each interval. For μ=600 and σ= 100, these intervals transform to X boundaries, (Fill in "Cell" column below.) Prior to sampling, it is presumed that the population mean μ is 600 with standard deviation σ of 100. As noted at the end of Lecture 14.1b, the eight z-intervals (-00, -1.15), [-1.15, -0.675), [-0.675, -0.32), [-0.32, 0), [0, 0.32), [0.32, 0.675), [0.675, 1.15) and [1.15, 0) have a probability of 1/8 each, i.e. the probability is uniform for each interval. For u 600 and σ= 100, these intervals transform to X boundaries, (Fill in "Cell" column below.) The observed counts are ... (Fill in "Observed" column below.) Next step: We need to transform these X boundaries into estimated (expected) cell probabilities, л, (a,ô), using the maximum likelihood estimates û and . Back in Lecture 6.2, we derived formulas for both of these. Cell Observed n₁ Estimated Expected =ηπ (0,6) (0-E)² (n; (O-E) (0-E)² E ‚¯në₁(û‚ô))² nл (Û‚ô) x² = Σ (0-E)² E IMPORTANT: The symbol x² is a notation! Do not square the sum in the last column. critical values: conclusion: