1. Problem 1: Hypothesis test for population proportion. We want to test if the population proportion $p$ of households loyal to brand W is greater than 0.48. 2. State hypotheses: $$H_0: p \leq 0.48$$ $$H_a: p > 0.48$$ 3. Given data: Sample size $n=100$, sample proportion $\hat{p} = \frac{50}{100} = 0.5$, significance level $\alpha=0.01$. 4. Test statistic formula for proportion: $$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$ where $p_0=0.48$. 5. Calculate standard error: $$SE = \sqrt{\frac{0.48 \times (1-0.48)}{100}} = \sqrt{\frac{0.48 \times 0.52}{100}} = \sqrt{0.002496} = 0.04996$$ 6. Calculate $z$: $$z = \frac{0.5 - 0.48}{0.04996} = \frac{0.02}{0.04996} = 0.4003$$ 7. Critical value for $\alpha=0.01$ (one-tailed) is $z_{0.01} = 2.33$. 8. Since $z=0.4003 < 2.33$, we fail to reject $H_0$. 9. Conclusion: There is insufficient evidence at 1% significance level to conclude that the population proportion is greater than 0.48. --- 10. Problem 2a(i): Number of ways to select first, second, and third place winners from 10 photos. 11. Since order matters and no repetition, use permutation: $$P(10,3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} = 10 \times 9 \times 8 = 720$$ --- 12. Problem 2a(ii): Number of ways to select 4 trainees from 12 applicants. 13. Since order does not matter, use combination: $$C(12,4) = \frac{12!}{4! \times 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$$ --- 14. Problem 3(i): Calculate product moment correlation coefficient $r$ for given data. 15. Data: $X = [8,10,6,4,5,6]$ $Y = [89,92,84,75,80,80]$ 16. Calculate means: $$\bar{X} = \frac{8+10+6+4+5+6}{6} = \frac{39}{6} = 6.5$$ $$\bar{Y} = \frac{89+92+84+75+80+80}{6} = \frac{500}{6} = 83.33$$ 17. Calculate numerator and denominators for $r$: $$\sum (X_i - \bar{X})(Y_i - \bar{Y}) = (8-6.5)(89-83.33) + (10-6.5)(92-83.33) + (6-6.5)(84-83.33) + (4-6.5)(75-83.33) + (5-6.5)(80-83.33) + (6-6.5)(80-83.33)$$ $$= 1.5 \times 5.67 + 3.5 \times 8.67 + (-0.5) \times 0.67 + (-2.5) \times (-8.33) + (-1.5) \times (-3.33) + (-0.5) \times (-3.33)$$ $$= 8.505 + 30.345 - 0.335 + 20.825 + 4.995 + 1.665 = 65.0$$ $$\sum (X_i - \bar{X})^2 = (1.5)^2 + (3.5)^2 + (-0.5)^2 + (-2.5)^2 + (-1.5)^2 + (-0.5)^2 = 2.25 + 12.25 + 0.25 + 6.25 + 2.25 + 0.25 = 23.5$$ $$\sum (Y_i - \bar{Y})^2 = (5.67)^2 + (8.67)^2 + (0.67)^2 + (-8.33)^2 + (-3.33)^2 + (-3.33)^2 = 32.15 + 75.17 + 0.45 + 69.39 + 11.09 + 11.09 = 199.34$$ 18. Calculate $r$: $$r = \frac{65.0}{\sqrt{23.5 \times 199.34}} = \frac{65.0}{\sqrt{4687.99}} = \frac{65.0}{68.47} = 0.95$$ 19. Interpretation: $r=0.95$ indicates a very strong positive linear relationship between studying time and test score. --- 20. Problem 3(ii): Calculate Spearman's rank correlation coefficient $r_s$. 21. Rank the data: $X$: 4(1), 5(2), 6(3.5), 6(3.5), 8(5), 10(6) $Y$: 75(1), 80(2.5), 80(2.5), 84(4), 89(5), 92(6) 22. Calculate difference in ranks $d_i$ and $d_i^2$: |X rank|Y rank|$d_i$|$d_i^2$| |---|---|---|---| |5|5|0|0| |6|6|0|0| |3.5|4|-0.5|0.25| |1|1|0|0| |2|2.5|-0.5|0.25| |3.5|2.5|1|1| Sum of $d_i^2 = 0 + 0 + 0.25 + 0 + 0.25 + 1 = 1.5$ 23. Spearman's formula: $$r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 -1)} = 1 - \frac{6 \times 1.5}{6(36-1)} = 1 - \frac{9}{210} = 1 - 0.0429 = 0.9571$$ 24. Interpretation: $r_s=0.9571$ also shows a very strong positive correlation. --- 25. Problem 3(iii): Find least squares regression equation $\hat{Y} = a + bX$. 26. Calculate slope $b$: $$b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} = \frac{65.0}{23.5} = 2.77$$ 27. Calculate intercept $a$: $$a = \bar{Y} - b \bar{X} = 83.33 - 2.77 \times 6.5 = 83.33 - 18.01 = 65.32$$ 28. Regression equation: $$\hat{Y} = 65.32 + 2.77X$$ --- 29. Problem 3(iv): Estimate test score for $X=7$ hours. 30. Substitute $X=7$: $$\hat{Y} = 65.32 + 2.77 \times 7 = 65.32 + 19.39 = 84.71$$ Final answers: - Hypothesis test: Fail to reject $H_0$, insufficient evidence $p>0.48$. - Number of ways winners formed: 720. - Number of ways trainees formed: 495. - Product moment correlation coefficient $r=0.95$ (strong positive). - Spearman's rank correlation coefficient $r_s=0.9571$ (strong positive). - Regression equation: $\hat{Y} = 65.32 + 2.77X$. - Estimated score for 7 hours: 84.71.