1. **State the problem:** We are given data on the number of advertisements (x) and the number of houses sold (y). We need to analyze the relationship between these variables. 2. **Type of variable for houses sold:** The number of houses sold is a quantitative discrete variable because it counts the number of houses sold. 3. **Calculate the product moment correlation coefficient (r):** The formula for $r$ is: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ Given data: $x = [25, 45, 15, 40, 35, 6, 24, 12, 32, 21]$ $y = [18, 30, 13, 25, 18, 10, 18, 13, 20, 16]$ Calculate sums: $\sum x = 25+45+15+40+35+6+24+12+32+21 = 255$ $\sum y = 18+30+13+25+18+10+18+13+20+16 = 181$ $\sum xy = 25\times18 + 45\times30 + 15\times13 + 40\times25 + 35\times18 + 6\times10 + 24\times18 + 12\times13 + 32\times20 + 21\times16 = 25\times18+45\times30+15\times13+40\times25+35\times18+6\times10+24\times18+12\times13+32\times20+21\times16 = 450 + 1350 + 195 + 1000 + 630 + 60 + 432 + 156 + 640 + 336 = 5249$ $\sum x^2 = 25^2 + 45^2 + 15^2 + 40^2 + 35^2 + 6^2 + 24^2 + 12^2 + 32^2 + 21^2 = 625 + 2025 + 225 + 1600 + 1225 + 36 + 576 + 144 + 1024 + 441 = 7497$ $\sum y^2 = 18^2 + 30^2 + 13^2 + 25^2 + 18^2 + 10^2 + 18^2 + 13^2 + 20^2 + 16^2 = 324 + 900 + 169 + 625 + 324 + 100 + 324 + 169 + 400 + 256 = 3591$ Number of data points $n=10$ Calculate numerator: $$n\sum xy - \sum x \sum y = 10 \times 5249 - 255 \times 181 = 52490 - 46155 = 6335$$ Calculate denominator: $$\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)} = \sqrt{(10 \times 7497 - 255^2)(10 \times 3591 - 181^2)} = \sqrt{(74970 - 65025)(35910 - 32761)} = \sqrt{9945 \times 3149}$$ Calculate: $$\sqrt{9945 \times 3149} = \sqrt{31322405} \approx 5595.56$$ Therefore, $$r = \frac{6335}{5595.56} \approx 1.1327$$ Since $r$ cannot be greater than 1, re-check calculations: Recalculate $\sum xy$: $25\times18=450$ $45\times30=1350$ $15\times13=195$ $40\times25=1000$ $35\times18=630$ $6\times10=60$ $24\times18=432$ $12\times13=156$ $32\times20=640$ $21\times16=336$ Sum: $450+1350+195+1000+630+60+432+156+640+336=5249$ correct. Recalculate denominator terms: $n\sum x^2 - (\sum x)^2 = 10 \times 7497 - 255^2 = 74970 - 65025 = 9945$ $n\sum y^2 - (\sum y)^2 = 10 \times 3591 - 181^2 = 35910 - 32761 = 3149$ Multiply: $9945 \times 3149 = 31322405$ Square root: $\sqrt{31322405} \approx 5595.56$ Calculate $r$ again: $$r = \frac{6335}{5595.56} \approx 1.1327$$ This is impossible; $r$ must be between -1 and 1. Check sums of $x$ and $y$ again: $\sum x = 25+45+15+40+35+6+24+12+32+21 = 255$ $\sum y = 18+30+13+25+18+10+18+13+20+16 = 181$ Check $\sum xy$ again: $450 + 1350 + 195 + 1000 + 630 + 60 + 432 + 156 + 640 + 336 = 5249$ Check $\sum x^2$: $625 + 2025 + 225 + 1600 + 1225 + 36 + 576 + 144 + 1024 + 441 = 7497$ Check $\sum y^2$: $324 + 900 + 169 + 625 + 324 + 100 + 324 + 169 + 400 + 256 = 3591$ All sums are correct. Recalculate numerator: $$n\sum xy - \sum x \sum y = 10 \times 5249 - 255 \times 181 = 52490 - 46155 = 6335$$ Recalculate denominator: $$\sqrt{(10 \times 7497 - 255^2)(10 \times 3591 - 181^2)} = \sqrt{9945 \times 3149} = \sqrt{31322405} \approx 5595.56$$ Calculate $r$: $$r = \frac{6335}{5595.56} \approx 1.1327$$ Since this is impossible, the error is likely in the calculation of $\sum xy$ or sums. Recalculate $\sum xy$ carefully: $25\times18=450$ $45\times30=1350$ $15\times13=195$ $40\times25=1000$ $35\times18=630$ $6\times10=60$ $24\times18=432$ $12\times13=156$ $32\times20=640$ $21\times16=336$ Sum: $450+1350=1800$, $1800+195=1995$, $1995+1000=2995$, $2995+630=3625$, $3625+60=3685$, $3685+432=4117$, $4117+156=4273$, $4273+640=4913$, $4913+336=5249$ Sum is correct. Check $\sum x \sum y = 255 \times 181 = 46155$ Calculate numerator again: $10 \times 5249 = 52490$ $52490 - 46155 = 6335$ Denominator: $10 \times 7497 = 74970$ $255^2 = 65025$ $74970 - 65025 = 9945$ $10 \times 3591 = 35910$ $181^2 = 32761$ $35910 - 32761 = 3149$ Multiply: $9945 \times 3149 = 31322405$ Square root: $\sqrt{31322405} \approx 5595.56$ Calculate $r$: $6335 / 5595.56 = 1.1327$ This is impossible; $r$ must be between -1 and 1. Check if data points are correct or if any data entry error. Assuming data is correct, the error is in the formula application. Formula for $r$ is correct. Try calculating $r$ using the alternative formula: $$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$ Calculate means: $\bar{x} = \frac{255}{10} = 25.5$ $\bar{y} = \frac{181}{10} = 18.1$ Calculate $\sum (x_i - \bar{x})(y_i - \bar{y})$: | x | y | $x_i - \bar{x}$ | $y_i - \bar{y}$ | Product | |---|---|----------------|----------------|---------| |25 |18 | -0.5 | -0.1 | 0.05 | |45 |30 | 19.5 | 11.9 | 232.05 | |15 |13 | -10.5 | -5.1 | 53.55 | |40 |25 | 14.5 | 6.9 | 100.05 | |35 |18 | 9.5 | -0.1 | -0.95 | |6 |10 | -19.5 | -8.1 | 157.95 | |24 |18 | -1.5 | -0.1 | 0.15 | |12 |13 | -13.5 | -5.1 | 68.85 | |32 |20 | 6.5 | 1.9 | 12.35 | |21 |16 | -4.5 | -2.1 | 9.45 | Sum of products: $0.05 + 232.05 + 53.55 + 100.05 - 0.95 + 157.95 + 0.15 + 68.85 + 12.35 + 9.45 = 633.5$ Calculate $\sum (x_i - \bar{x})^2$: $(-0.5)^2 + 19.5^2 + (-10.5)^2 + 14.5^2 + 9.5^2 + (-19.5)^2 + (-1.5)^2 + (-13.5)^2 + 6.5^2 + (-4.5)^2 = 0.25 + 380.25 + 110.25 + 210.25 + 90.25 + 380.25 + 2.25 + 182.25 + 42.25 + 20.25 = 1418.5$ Calculate $\sum (y_i - \bar{y})^2$: $(-0.1)^2 + 11.9^2 + (-5.1)^2 + 6.9^2 + (-0.1)^2 + (-8.1)^2 + (-0.1)^2 + (-5.1)^2 + 1.9^2 + (-2.1)^2 = 0.01 + 141.61 + 26.01 + 47.61 + 0.01 + 65.61 + 0.01 + 26.01 + 3.61 + 4.41 = 314.9$ Calculate denominator: $$\sqrt{1418.5 \times 314.9} = \sqrt{446799.65} \approx 668.5$$ Calculate $r$: $$r = \frac{633.5}{668.5} \approx 0.9477$$ 4. **Interpretation:** The correlation coefficient $r \approx 0.9477$ indicates a very strong positive linear relationship between the number of advertisements and houses sold. 5. **Find the linear regression equation $y = a + bx$:** Slope $b$ is given by: $$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{633.5}{1418.5} \approx 0.4465$$ Intercept $a$ is: $$a = \bar{y} - b \bar{x} = 18.1 - 0.4465 \times 25.5 = 18.1 - 11.38 = 6.72$$ Therefore, the regression equation is: $$y = 6.7200 + 0.4465x$$ 6. **Interpret coefficients:** - Intercept $a = 6.7200$: When there are zero advertisements, approximately 6.72 houses are expected to be sold. - Gradient $b = 0.4465$: For each additional advertisement, the number of houses sold increases by about 0.4465 houses. 7. **Estimate houses sold for 20 advertisements:** Substitute $x=20$ into the regression equation: $$y = 6.7200 + 0.4465 \times 20 = 6.7200 + 8.930 = 15.6500$$ So, approximately 15.65 houses are expected to be sold when 20 advertisements are made.