To find the estimated value of $y$ when $x = 43$ using the regression equation $\hat{y} = b_0 + b_1x$, we need to calculate the regression coefficients $b_0$ (intercept) and $b_1$ (slope).

Let's start by calculating the slope $b_1$ and the intercept $b_0$.

Step 1: Calculate the necessary statistics

We are given the following data:

Price ($x$)	Number of Bids ($y$)
20	3
39	5
43	8
44	9
47	10

Calculating the means:

The formula for the mean of $x$ is: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ The formula for the mean of $y$ is: $\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i$

$\bar{x} = \frac{20 + 39 + 43 + 44 + 47}{5} = \frac{193}{5} = 38.6$ $\bar{y} = \frac{3 + 5 + 8 + 9 + 10}{5} = \frac{35}{5} = 7$

Calculating the slope ($b_1$) and intercept ($b_0$):

The formula for the slope $b_1$ is: $b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

The formula for the intercept $b_0$ is: $b_0 = \bar{y} - b_1 \bar{x}$

Let me compute those. numpy as np

Data

x = np.array([20, 39, 43, 44, 47]) y = np.array([3, 5, 8, 9, 10])

Means

x_bar = np.mean(x) y_bar = np.mean(y)

Calculating b1 (slope)

numerator = np.sum((x - x_bar) * (y - y_bar)) denominator = np.sum((x - x_bar) ** 2) b1 = numerator / denominator

Calculating b0 (intercept)

b0 = y_bar - b1 * x_bar

b1, b0The calculated values for the regression coefficients are:

• Slope $b_1$ ≈ 0.245
• Intercept $b_0$ ≈ -2.459

Step 2: Calculate the estimated value of $y$ when $x = 43$

Now, we can use the regression equation $\hat{y} = b_0 + b_1x$ to find the estimated number of bids when the price is $43.

$\hat{y} = -2.459 + 0.245 \times 43$ $\hat{y} = -2.459 + 10.535$ $\hat{y} ≈ 8.076$

So, the estimated number of bids when the price is $43 is approximately 8.076.

Would you like further details or have any questions?

Related Questions:

1. How do you interpret the slope and intercept of the regression line?
2. How do you calculate the correlation coefficient for this data?
3. What is the significance of statistical tests for correlation in regression?
4. How can you calculate the standard error of the regression?
5. What would happen if the price was higher or lower than the given range in the dataset?

Tip:

To improve the accuracy of predictions in real-world applications, always check for the statistical significance of the regression model by evaluating the correlation coefficient and conducting hypothesis tests.