1. **Stating the problem:** We have data on the number of hotspots (independent variable $x$) and the number of internet users (dependent variable $y$) over 5 months. We want to: a. Find the linear regression equation $y = a + bx$. b. Calculate the correlation coefficient $r$. c. Predict $y$ when $x=30$. 2. **Data:** \begin{array}{c|c|c} \text{Month} & x=\text{Hotspots} & y=\text{Users} \\ \hline 1 & 5 & 15 \\ 2 & 8 & 40 \\ 3 & 15 & 110 \\ 4 & 20 & 160 \\ 5 & 25 & 220 \end{array} 3. **Calculate sums and means:** $$\sum x = 5+8+15+20+25=73$$ $$\sum y = 15+40+110+160+220=545$$ $$\sum x^2 = 5^2+8^2+15^2+20^2+25^2=25+64+225+400+625=1339$$ $$\sum y^2 = 15^2+40^2+110^2+160^2+220^2=225+1600+12100+25600+48400=87925$$ $$\sum xy = 5\times15 + 8\times40 + 15\times110 + 20\times160 + 25\times220 = 75 + 320 + 1650 + 3200 + 5500 = 10745$$ $$n=5$$ $$\bar{x} = \frac{73}{5} = 14.6$$ $$\bar{y} = \frac{545}{5} = 109$$ 4. **Calculate slope $b$ of regression line:** $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{5\times10745 - 73\times545}{5\times1339 - 73^2} = \frac{53725 - 39785}{6695 - 5329} = \frac{13940}{1366} \approx 10.20$$ 5. **Calculate intercept $a$:** $$a = \bar{y} - b\bar{x} = 109 - 10.20 \times 14.6 = 109 - 148.92 = -39.92$$ 6. **Regression equation:** $$y = -39.92 + 10.20x$$ 7. **Calculate correlation coefficient $r$:** $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} = \frac{53725 - 39785}{\sqrt{(6695 - 5329)(439625 - 297025)}} = \frac{13940}{\sqrt{1366 \times 142600}} = \frac{13940}{\sqrt{194831600}} = \frac{13940}{13956.5} \approx 0.9988$$ 8. **Prediction for $x=30$:** $$y = -39.92 + 10.20 \times 30 = -39.92 + 306 = 266.08$$ **Final answers:** - Regression line: $y = -39.92 + 10.20x$ - Correlation coefficient: $r \approx 0.999$ - Predicted users for 30 hotspots: $266$