1. **State the problem:** We have prices of 8 items in 2001 and 2021. We need to find the mean and standard deviation for both years, then calculate the covariance and correlation coefficient between the prices in the two years. 2. **List the prices:** 2001 prices $X = [2.70, 1.50, 2.10, 2.70, 1.50, 4.10, 3.50, 3.50]$ 2021 prices $Y = [3.10, 2.00, 2.70, 2.95, 2.95, 4.80, 4.20, 4.50]$ 3. **Calculate the mean of each year:** $$\bar{X} = \frac{2.70+1.50+2.10+2.70+1.50+4.10+3.50+3.50}{8} = \frac{21.60}{8} = 2.70$$ $$\bar{Y} = \frac{3.10+2.00+2.70+2.95+2.95+4.80+4.20+4.50}{8} = \frac{27.20}{8} = 3.40$$ 4. **Calculate the standard deviation for each year:** Standard deviation formula: $$s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}$$ For 2001: $$\sum (X_i - \bar{X})^2 = (2.70-2.70)^2 + (1.50-2.70)^2 + (2.10-2.70)^2 + (2.70-2.70)^2 + (1.50-2.70)^2 + (4.10-2.70)^2 + (3.50-2.70)^2 + (3.50-2.70)^2$$ $$= 0 + 1.44 + 0.36 + 0 + 1.44 + 1.96 + 0.64 + 0.64 = 6.48$$ $$s_X = \sqrt{\frac{6.48}{7}} = \sqrt{0.9257} \approx 0.962$$ For 2021: $$\sum (Y_i - \bar{Y})^2 = (3.10-3.40)^2 + (2.00-3.40)^2 + (2.70-3.40)^2 + (2.95-3.40)^2 + (2.95-3.40)^2 + (4.80-3.40)^2 + (4.20-3.40)^2 + (4.50-3.40)^2$$ $$= 0.09 + 1.96 + 0.49 + 0.2025 + 0.2025 + 1.96 + 0.64 + 1.21 = 6.6495$$ $$s_Y = \sqrt{\frac{6.6495}{7}} = \sqrt{0.95} \approx 0.975$$ 5. **Calculate the covariance between $X$ and $Y$:** Covariance formula: $$\text{cov}(X,Y) = \frac{1}{n-1} \sum (X_i - \bar{X})(Y_i - \bar{Y})$$ Calculate each product: $$(2.70 - 2.70)(3.10 - 3.40) = 0\times(-0.30) = 0$$ $$(1.50 - 2.70)(2.00 - 3.40) = (-1.20)(-1.40) = 1.68$$ $$(2.10 - 2.70)(2.70 - 3.40) = (-0.60)(-0.70) = 0.42$$ $$(2.70 - 2.70)(2.95 - 3.40) = 0\times(-0.45) = 0$$ $$(1.50 - 2.70)(2.95 - 3.40) = (-1.20)(-0.45) = 0.54$$ $$(4.10 - 2.70)(4.80 - 3.40) = 1.40\times1.40 = 1.96$$ $$(3.50 - 2.70)(4.20 - 3.40) = 0.80\times0.80 = 0.64$$ $$(3.50 - 2.70)(4.50 - 3.40) = 0.80\times1.10 = 0.88$$ Sum of products = $0 + 1.68 + 0.42 + 0 + 0.54 + 1.96 + 0.64 + 0.88 = 6.12$ $${\rm cov}(X,Y) = \frac{6.12}{7} \approx 0.874$$ 6. **Calculate the correlation coefficient $r$:** $$r = \frac{{\rm cov}(X,Y)}{s_X s_Y} = \frac{0.874}{0.962 \times 0.975} = \frac{0.874}{0.938} \approx 0.932$$ **Final answers:** - Mean 2001 prices: $2.70$ - Standard deviation 2001: $0.962$ - Mean 2021 prices: $3.40$ - Standard deviation 2021: $0.975$ - Covariance: $0.874$ - Correlation coefficient: $0.932$