Descriptive Statistics

Part 1: Descriptive Statistics Definitions • Mean (Average): Pn i=1 x̄ = xi n • Median: The middle value when the data is sorted. • Variance (Sample): 2 Pn − x̄)2 n−1 i=1 (xi s = • Standard Deviation (Sample): √ s= s2 Problem 1: Descriptive Statistics Given the dataset {5, 7, 9, 11, 15}, calculate: 1. Mean 2. Median 3. Variance 4. Standard Deviation Solution (Step-by-Step): 1. Mean: x̄ = 5 + 7 + 9 + 11 + 15 47 = = 9.4 5 5 2. Median: - Sorted data: {5, 7, 9, 11, 15} - Middle value: 9 (since it is the third value in the sorted dataset). Median = 9 1 3. Variance: - Calculate deviations: (xi − x̄)2 : (5 − 9.4)2 = 19.36, (7 − 9.4)2 = 5.76, (11 − 9.4)2 = 2.56, (9 − 9.4)2 = 0.16, (15 − 9.4)2 = 31.36 - Sum of squared deviations: X (xi − x̄)2 = 19.36 + 5.76 + 0.16 + 2.56 + 31.36 = 59.2 - Variance: P (xi − x̄)2 59.2 s = = = 14.8 n−1 4 2 4. Standard Deviation: √ s= s2 = √ 14.8 ≈ 3.85 Part 2: Probability Distributions Problem 2: Probability A fair coin is flipped 10 times. Calculate: 1. The probability of getting exactly 6 heads. 2. The probability of getting at least 8 heads. Solution (Step-by-Step): 1. Exactly 6 heads (Binomial Distribution): - Formula:   n k P (X = k) = p (1 − p)n−k k - Substitute values: n = 10, k = 6, p = 0.5:   10 P (X = 6) = (0.5)6 (0.5)4 6
- Calculate 10 : 6   10! 10 10 × 9 × 8 × 7 = = = 210 6 6!(10 − 6)! 4×3×2×1 - Probability: P (X = 6) = 210(0.5)10 = 210 × 0.0009765625 = 0.205 2 2. At least 8 heads: - Use P (X ≥ 8) = P (X = 8)+P (X = 9)+P (X = 10). - Calculate each probability:     10 10 8 2 P (X = 8) = (0.5) (0.5) = (0.5)10 8 8     10 10 10 × 9 = = = 45, P (X = 8) = 45 × 0.0009765625 = 0.0439 8 2 2×1 Similarly: P (X = 9) = 10 × (0.5)10 = 0.0098, P (X = 10) = 1 × (0.5)10 = 0.000976 - Add probabilities: P (X ≥ 8) = 0.0439 + 0.0098 + 0.000976 = 0.0547 Part 3: Hypothesis Testing Problem 3: Testing a Mean A factory claims that the mean weight of its packages is 500g. A sample of 30 packages has a mean weight of 495g with a standard deviation of 10g. Test the claim at α = 0.05. Solution (Step-by-Step): 1. Set hypotheses: H0 : µ = 500 (Null Hypothesis) Ha : µ ̸= 500 (Alternative Hypothesis, two-tailed test) 2. Calculate test statistic: - Formula: z= x̄ − µ √s n - Substitute values: x̄ = 495, µ = 500, s = 10, n = 30: z= 495 − 500 √10 30 = −5 = −2.74 1.8257 3. Determine critical value: - For α = 0.05, the critical z-value for a two-tailed test is −1.96 and 1.96. 4. Conclusion: - Since z = −2.74 < −1.96, reject H0 . - There is sufficient evidence to suggest the mean weight is not 500g. 3