Let's break down how to solve this problem step by step, given the data provided:

Given Data:

• High temperature, $x$ (in °C): 12, 18, 23, 7, 29, 12, 23, 3, 15
• Amount of hot chocolate sold, $y$ (in liters): 10, 7, 11, 16, 5, 15, 8, 18, 12
• Product, $xy$: 120, 126, 253, 112, 145, 180, 184, 54, 180

Step 1: Calculate Required Sums

Let's calculate the following values:

1. $n$: The sample size (number of data points), which is 9.
2. $\sum x$: Sum of all $x$-values.
3. $\sum y$: Sum of all $y$-values.
4. $\sum x^2$: Sum of squares of $x$-values.
5. $\sum y^2$: Sum of squares of $y$-values.
6. $\sum xy$: Sum of the products $xy$.

Calculations:

1. $n = 9$

2. $\sum x = 12 + 18 + 23 + 7 + 29 + 12 + 23 + 3 + 15 = 142$

3. $\sum y = 10 + 7 + 11 + 16 + 5 + 15 + 8 + 18 + 12 = 102$

4. $\sum x^2 = 12^2 + 18^2 + 23^2 + 7^2 + 29^2 + 12^2 + 23^2 + 3^2 + 15^2$ $= 144 + 324 + 529 + 49 + 841 + 144 + 529 + 9 + 225 = 2794$

5. $\sum y^2 = 10^2 + 7^2 + 11^2 + 16^2 + 5^2 + 15^2 + 8^2 + 18^2 + 12^2$ $= 100 + 49 + 121 + 256 + 25 + 225 + 64 + 324 + 144 = 1308$

6. $\sum xy = 120 + 126 + 253 + 112 + 145 + 180 + 184 + 54 + 180 = 1354$

Step 2: Calculate Sample Means

Calculate the means of $x$ and $y$:

• $\bar{x} = \frac{\sum x}{n} = \frac{142}{9} \approx 15.778$
• $\bar{y} = \frac{\sum y}{n} = \frac{102}{9} \approx 11.333$

Step 3: Calculate the Correlation Coefficient $r$

The formula for the correlation coefficient $r$ 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]}} ] Plug in the values: [ r = \frac{9(1354) - (142)(102)}{\sqrt{[9(2794) - 142^2][9(1308) - 102^2]}} ] [ r = \frac{12186 - 14484}{\sqrt{[25146 - 20164][11772 - 10404]}} ] $r = \frac{-1298}{\sqrt{4982 \times 1368}} \approx \frac{-1298}{\sqrt{6815856}} \approx \frac{-1298}{2611.49} \approx -0.497$

Step 4: Calculate the Slope $b_1$

The formula for the slope $b_1$ is: $b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$ $b_1 = \frac{9(1354) - (142)(102)}{9(2794) - 142^2}$ $b_1 = \frac{12186 - 14484}{25146 - 20164} \approx \frac{-1298}{4982} \approx -0.261$

Step 5: Calculate the Y-Intercept $b_0$

The formula for the y-intercept $b_0$ is: $b_0 = \bar{y} - b_1 \bar{x}$ $b_0 = 11.333 - (-0.261)(15.778) \approx 11.333 + 4.114 \approx 15.447$

Step 6: Write the Regression Equation

The regression equation is: $y = -0.261x + 15.447$

Step 7: Predict Hot Chocolate Sales for 19°C

Substitute $x = 19$ into the regression equation: $y = -0.261(19) + 15.447 \approx -4.959 + 15.447 \approx 10.488$ Rounding to the nearest whole number, the predicted amount of hot chocolate sold is approximately 10 liters.

---

Would you like more details or have any questions?

Here are five related questions:

1. How does the correlation coefficient interpret the relationship between temperature and hot chocolate sales?
2. How do you determine if the regression model is a good fit for the data?
3. What is the significance of a negative slope in the regression equation?
4. How would adding more data points affect the correlation and regression line?
5. What other factors might influence hot chocolate sales besides temperature?

Tip: Always plot the scatter plot first to visually inspect the relationship between variables before calculating correlation or regression.