To estimate the regression function $Y = a + bX$, where $Y$ is the dependent variable (number of used cars sold in thousands) and $X$ is the independent variable (price in thousands), we will calculate the coefficients $a$ (intercept) and $b$ (slope) using the least squares method.

Step 1: Gather the data

The data points are:

$X$	$Y$
12	213
18	210
14	205
20	203

Step 2: Apply the least squares formulas

1. Slope $b$: $b = \frac{n\sum(XY) - \sum(X)\sum(Y)}{n\sum(X^2) - (\sum(X))^2}$

2. Intercept $a$: $a = \frac{\sum(Y) - b\sum(X)}{n}$

Where:

• $n$ is the number of data points (4 in this case),
• $\sum(XY)$ is the sum of the product of $X$ and $Y$,
• $\sum(X^2)$ is the sum of squares of $X$,
• $\sum(X)$ and $\sum(Y)$ are the sums of $X$ and $Y$, respectively.

Step 3: Perform the calculations

Let me compute these step by step. numpy as np

Data points

X = np.array([12, 18, 14, 20]) # Prices in thousands Y = np.array([213, 210, 205, 203]) # Number of cars sold in thousands

Number of data points

n = len(X)

Calculate sums and intermediate terms

sum_X = np.sum(X) sum_Y = np.sum(Y) sum_XY = np.sum(X * Y) sum_X2 = np.sum(X ** 2)

Calculate slope (b) and intercept (a)

b = (n * sum_XY - sum_X * sum_Y) / (n * sum_X2 - sum_X ** 2) a = (sum_Y - b * sum_X) / n

a, bThe regression equation estimated from the given data is:

$Y = 219.75 - 0.75X$

Interpretation:

1. Intercept ($a = 219.75$): If the price of used cars ($X$) were zero, the estimated number of used cars sold ($Y$) would be 219.75 thousand.
2. Slope ($b = -0.75$): For every 1 thousand dollar increase in the price of used cars, the number of cars sold decreases by 0.75 thousand.

Let me know if you'd like to discuss implications, perform additional analysis, or verify predictions using this equation!

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Related Questions:

1. What is the predicted number of cars sold when prices are $16,000?
2. How well does this regression model fit the data?
3. Can you calculate the residuals for each data point using this model?
4. How would the regression change if an outlier price like $50,000 was added?
5. What assumptions are made in linear regression, and do they hold for this data?

Tip:

In regression analysis, always plot the data and the regression line to visually inspect the model fit.