1. **Problem:** Find the frequency corresponding to the cumulative frequency (cf) 174 from the given distribution. 2. **Given Data:** Classes Midpoint: 400, 389, 365, 322, 270, 200, 174, 11, 10 Frequency (f): 22, __, __, __, __, __, __, __, __ Cumulative Frequency (cf): __, __, __, __, __, __, 174, __, __ 3. **Step:** The cumulative frequency (cf) is the sum of frequencies up to a certain class. 4. Since cf 174 is given, and frequency for 174 is 11, the frequency at cf 174 is 11. **Answer:** 11 --- 2. **Problem:** Find the class interval corresponding to the midpoint 55. 3. **Step:** The midpoint of a class interval is calculated as $\frac{\text{Lower limit} + \text{Upper limit}}{2}$. 4. To find the class interval for midpoint 55, solve: $$\frac{L + U}{2} = 55$$ 5. Multiply both sides by 2: $$L + U = 110$$ 6. Since class intervals are usually equal width, assume width $w$ and write: $$U = L + w$$ 7. Substitute: $$L + (L + w) = 110 \Rightarrow 2L + w = 110$$ 8. Without width given, assume width 10 (common in statistics): $$2L + 10 = 110 \Rightarrow 2L = 100 \Rightarrow L = 50$$ 9. Then upper limit: $$U = 50 + 10 = 60$$ **Answer:** (50-60) --- 3. **Problem:** Interpretation of the data distribution. 4. Given options: - a. peaked in the center - b. more high scorers - c. more low scorers 5. Without exact data, but given frequencies and midpoints, if frequencies are higher at lower midpoints, then more low scorers. **Answer:** c. more low scorers --- 4. **Problem:** Identify the type of distribution from given classes and frequencies. 5. Given classes and frequencies: 95-99:1, 90-94:0, 85-89:6, 80-84:5, 75-79:5, 70-74:10, 65-69:2, 60-64:1 6. Observing frequencies, higher frequencies are at lower classes, tail towards higher classes. 7. This indicates a positively skewed distribution. **Answer:** b. positively skewed --- 5. **Problem:** Interpretation of the data performance. 6. Options: - a. The group is low performing - b. The group is high performing - c. The group has an average performance 7. Given more low scores and positive skew, group is low performing. **Answer:** a. The group is low performing --- 6. **Problem:** Measure of central tendency for platykurtic curve. 7. Platykurtic means flat distribution, median is less affected by extremes. **Answer:** b. Median --- 7. **Problem:** Measure of central tendency for positively skewed distribution. 8. For positive skew, median is preferred. **Answer:** a. Median --- 8. **Problem:** Find mean of data: 75, 75, 86, 85, 88, 89, 90, 91, 92 9. Sum data: $$75 + 75 + 86 + 85 + 88 + 89 + 90 + 91 + 92 = 771$$ 10. Number of data points: 9 11. Mean: $$\frac{771}{9} = 85.6667$$ 12. Rounded to two decimals: $$85.67$$ **Answer:** 85.67 --- 9. **Problem:** Find Q (quartile) of same data. 10. Sort data: 75, 75, 85, 86, 88, 89, 90, 91, 92 11. Median (Q2) is 5th value: 88 12. Q1 (lower quartile) is median of first 4 values: 75, 75, 85, 86 13. Q1: $$\frac{75 + 85}{2} = 80$$ 14. Q3 (upper quartile) median of last 4 values: 89, 90, 91, 92 15. Q3: $$\frac{90 + 91}{2} = 90.5$$ **Answer:** Median (Q2) = 88.00 --- 10. **Problem:** Compute sample standard deviation of data in 8. 11. Mean $\bar{x} = 85.67$ 12. Calculate squared deviations: $(75-85.67)^2 = 113.49$ $(75-85.67)^2 = 113.49$ $(86-85.67)^2 = 0.11$ $(85-85.67)^2 = 0.45$ $(88-85.67)^2 = 5.44$ $(89-85.67)^2 = 11.11$ $(90-85.67)^2 = 18.71$ $(91-85.67)^2 = 28.44$ $(92-85.67)^2 = 40.11$ 13. Sum squared deviations: $$113.49 + 113.49 + 0.11 + 0.45 + 5.44 + 11.11 + 18.71 + 28.44 + 40.11 = 331.35$$ 14. Sample variance: $$s^2 = \frac{331.35}{9-1} = \frac{331.35}{8} = 41.42$$ 15. Sample standard deviation: $$s = \sqrt{41.42} = 6.44$$ **Answer:** 6.44 --- 11. **Problem:** Compute population standard deviation for data: 70, 73, 85, 89, 92, 79, 82, 68, 78, 60 12. Mean: $$\bar{x} = \frac{70+73+85+89+92+79+82+68+78+60}{10} = \frac{776}{10} = 77.6$$ 13. Squared deviations: $(70-77.6)^2=57.76$ $(73-77.6)^2=21.16$ $(85-77.6)^2=54.76$ $(89-77.6)^2=129.96$ $(92-77.6)^2=207.36$ $(79-77.6)^2=1.96$ $(82-77.6)^2=19.36$ $(68-77.6)^2=92.16$ $(78-77.6)^2=0.16$ $(60-77.6)^2=313.6$ 14. Sum: $$57.76 + 21.16 + 54.76 + 129.96 + 207.36 + 1.96 + 19.36 + 92.16 + 0.16 + 313.6 = 898.24$$ 15. Population variance: $$\sigma^2 = \frac{898.24}{10} = 89.824$$ 16. Population standard deviation: $$\sigma = \sqrt{89.824} = 9.48$$ **Answer:** 9.48 --- 12. **Problem:** Given population 800, mean 78, SD 9, find number of scores above 89. 13. Calculate z-score for 89: $$z = \frac{89 - 78}{9} = \frac{11}{9} = 1.22$$ 14. Using standard normal table, P(Z > 1.22) = 0.1112 15. Number of scores above 89: $$800 \times 0.1112 = 88.96 \approx 89$$ **Answer:** 89 --- 13. **Problem:** Probability of score below 92. 14. Calculate z-score: $$z = \frac{92 - 78}{9} = \frac{14}{9} = 1.5556$$ 15. From z-table, P(Z < 1.5556) = 0.94060 **Answer:** 0.94060 --- 14. **Problem:** Probability between 72 and 89. 15. Calculate z-scores: $$z_{72} = \frac{72 - 78}{9} = -0.6667$$ $$z_{89} = 1.22$$ 16. From z-table: P(Z < 1.22) = 0.8888 P(Z < -0.6667) = 0.2525 17. Probability between: $$0.8888 - 0.2525 = 0.6363$$ **Answer:** 0.63630 --- 15. **Problem:** Probability of score below 72. 16. z-score: $$z = -0.6667$$ 17. From z-table: P(Z < -0.6667) = 0.2525 **Answer:** 0.25250 --- 16. **Problem:** Find score marking 80th percentile. 17. From z-table, z for 80th percentile is 0.8416 18. Use formula: $$X = \mu + z \times \sigma = 78 + 0.8416 \times 9 = 78 + 7.5744 = 85.57$$ **Answer:** 85.57 --- 17. **Problem:** Interpretation of 80th percentile score. 18. Options: - a. 20 percent of test takers are above this score - b. The mass of test takers is below the score - c. The score marks 80 percent of takers 19. By definition, 80th percentile means 80% scored below, 20% above. **Answer:** a. 20 percent of the test takers are above this score