To find the estimated value of $y$ when $x = 46$, we need to follow these steps:

Step 1: Compute the Regression Coefficients $b_0$ and $b_1$

The equation of the regression line is:

$\hat{y} = b_0 + b_1 x$

Where:

• $b_0$ is the y-intercept
• $b_1$ is the slope

Step 2: Calculate the Necessary Values

We need the following formulas to compute $b_0$ and $b_1$:

$b_1 = \frac{n \sum x y - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

$b_0 = \frac{\sum y - b_1 \sum x}{n}$

Where:

• $x$ = price in dollars
• $y$ = number of bids
• $n$ = number of data points

Data:

$x = [23, 34, 44, 46, 50]$ $y = [1, 2, 4, 9, 10]$

Step 3: Compute the Sums

Let's calculate the following sums:

• $\sum x = 23 + 34 + 44 + 46 + 50 = 197$
• $\sum y = 1 + 2 + 4 + 9 + 10 = 26$
• $\sum x^2 = 23^2 + 34^2 + 44^2 + 46^2 + 50^2 = 529 + 1156 + 1936 + 2116 + 2500 = 8237$
• $\sum xy = (23 \times 1) + (34 \times 2) + (44 \times 4) + (46 \times 9) + (50 \times 10) = 23 + 68 + 176 + 414 + 500 = 1181$

Step 4: Calculate $b_1$

Now, substitute these values into the formula for $b_1$:

$b_1 = \frac{5(1181) - (197)(26)}{5(8237) - (197)^2}$

$b_1 = \frac{5905 - 5122}{41185 - 38809}$

$b_1 = \frac{783}{2386} \approx 0.328$

Step 5: Calculate $b_0$

Now, substitute the values into the formula for $b_0$:

$b_0 = \frac{26 - (0.328)(197)}{5}$

$b_0 = \frac{26 - 64.576}{5}$

$b_0 = \frac{-38.576}{5} \approx -7.715$

Step 6: Estimate $\hat{y}$ when $x = 46$

Now that we have $b_0 \approx -7.715$ and $b_1 \approx 0.328$, we can estimate $\hat{y}$ for $x = 46$:

$\hat{y} = -7.715 + 0.328(46)$

$\hat{y} = -7.715 + 15.088$

$\hat{y} \approx 7.373$

Final Answer:

The estimated value of $y$ when $x = 46$ is approximately 7.373 (rounded to three decimal places).

Would you like more details on how I calculated these steps, or have any questions? Here are some related questions you might find useful:

1. How do you interpret the slope and y-intercept in a regression line?
2. What assumptions are made when using a regression model like this?
3. How do you calculate the correlation coefficient, and why is it important?
4. What does it mean if the regression model has a low R-squared value?
5. How can we check if the regression coefficients are statistically significant?

Tip: Always check the residuals (the differences between the observed and predicted values) to assess the fit of the regression model.