1. **Problem Statement:** You are given a population of 20 numbers and need to find the population mean, variance, and standard deviation. Then, take 5 random samples of size 5 each, compute their means, and analyze the sampling distribution of these sample means. 2. **Population Data:** $$5, 8, 10, 12, 15, 16, 18, 20, 22, 25, 26, 28, 30, 32, 34, 35, 38, 40, 42, 45$$ 3. **Formulas:** - Population mean: $$\mu = \frac{\sum x_i}{N}$$ where $N=20$. - Population variance: $$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$$ - Population standard deviation: $$\sigma = \sqrt{\sigma^2}$$ - Sample mean for each sample: $$\bar{x} = \frac{\sum x_i}{n}$$ where $n=5$. - Variance and standard deviation of sample means (sampling distribution): calculate variance and std dev of the 5 sample means. 4. **Step 1: Calculate Population Mean** $$\mu = \frac{5 + 8 + 10 + 12 + 15 + 16 + 18 + 20 + 22 + 25 + 26 + 28 + 30 + 32 + 34 + 35 + 38 + 40 + 42 + 45}{20}$$ Sum = 511 $$\mu = \frac{511}{20} = 25.55$$ 5. **Step 2: Calculate Population Variance** Calculate each $(x_i - \mu)^2$ and sum: $$(5-25.55)^2=420.30, (8-25.55)^2=309.30, (10-25.55)^2=240.80, (12-25.55)^2=182.70, (15-25.55)^2=111.30,$$ $$(16-25.55)^2=91.30, (18-25.55)^2=56.70, (20-25.55)^2=30.80, (22-25.55)^2=12.60, (25-25.55)^2=0.30,$$ $$(26-25.55)^2=0.20, (28-25.55)^2=6.00, (30-25.55)^2=19.80, (32-25.55)^2=41.90, (34-25.55)^2=71.90,$$ $$(35-25.55)^2=89.30, (38-25.55)^2=156.90, (40-25.55)^2=207.20, (42-25.55)^2=271.20, (45-25.55)^2=380.70$$ Sum of squares = 2438.50 $$\sigma^2 = \frac{2438.50}{20} = 121.925$$ 6. **Step 3: Calculate Population Standard Deviation** $$\sigma = \sqrt{121.925} \approx 11.04$$ 7. **Step 4: Take 5 Random Samples of Size 5** Example samples (randomly chosen): - Sample 1: 5, 12, 25, 30, 42 - Sample 2: 8, 16, 22, 35, 45 - Sample 3: 10, 18, 26, 32, 40 - Sample 4: 15, 20, 28, 34, 38 - Sample 5: 12, 25, 30, 38, 42 8. **Step 5: Calculate Sample Means** - Sample 1 mean: $$\frac{5+12+25+30+42}{5} = \frac{114}{5} = 22.8$$ - Sample 2 mean: $$\frac{8+16+22+35+45}{5} = \frac{126}{5} = 25.2$$ - Sample 3 mean: $$\frac{10+18+26+32+40}{5} = \frac{126}{5} = 25.2$$ - Sample 4 mean: $$\frac{15+20+28+34+38}{5} = \frac{135}{5} = 27.0$$ - Sample 5 mean: $$\frac{12+25+30+38+42}{5} = \frac{147}{5} = 29.4$$ 9. **Step 6: Calculate Variance and Standard Deviation of Sample Means** Sample means: 22.8, 25.2, 25.2, 27.0, 29.4 Mean of sample means: $$\bar{\bar{x}} = \frac{22.8 + 25.2 + 25.2 + 27.0 + 29.4}{5} = \frac{129.6}{5} = 25.92$$ Calculate variance: $$(22.8 - 25.92)^2 = 9.74, (25.2 - 25.92)^2 = 0.52, (25.2 - 25.92)^2 = 0.52,$$ $$(27.0 - 25.92)^2 = 1.17, (29.4 - 25.92)^2 = 12.07$$ Sum = 23.99 $$s^2 = \frac{23.99}{5} = 4.80$$ Standard deviation: $$s = \sqrt{4.80} \approx 2.19$$ 10. **Step 7: Answer Questions** I. Sample means are close but not exactly equal to the population mean 25.55. II. Not all sample means equal the population mean because samples vary randomly. III. When more groups add their sample means, the average of these means tends to approach the population mean (Law of Large Numbers). IV. Random sampling is useful because it provides a practical way to estimate population parameters without surveying everyone, saving time and resources. **Final answers:** - Population mean: 25.55 - Population variance: 121.93 - Population standard deviation: 11.04 - Sample means: 22.8, 25.2, 25.2, 27.0, 29.4 - Mean of sample means: 25.92 - Variance of sample means: 4.80 - Standard deviation of sample means: 2.19