The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x
, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Price in Dollars	Number of Bids
28	2
29	3
35	4
36	6
50	7  Find the estimated slope. Round your answer to three decimal places.

To find the estimated slope \( b_1 \) of the regression line \( \hat{y} = b_0 + b_1x \), we can use the formula:

\[
b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]

Where:
- \( x \) represents the list price of the items.
- \( y \) represents the number of bids.
- \( n \) is the number of data points (in this case, 5 items).

### Step 1: Organize the data
The given data is:

\[
\begin{array}{|c|c|}
\hline
\text{Price (x)} & \text{Bids (y)} \\
\hline
28 & 2 \\
29 & 3 \\
35 & 4 \\
36 & 6 \\
50 & 7 \\
\hline
\end{array}
\]

### Step 2: Compute the necessary sums
We need to calculate:
1. \( \sum x \) (sum of the prices),
2. \( \sum y \) (sum of the bids),
3. \( \sum xy \) (sum of the products of the corresponding x and y values),
4. \( \sum x^2 \) (sum of the squares of the prices).

Let's calculate these values.The computed values are:

\[
\sum x = 178, \quad \sum y = 22, \quad \sum xy = 849, \quad \sum x^2 = 6646, \quad n = 5
\]

### Step 3: Apply the slope formula

Now, we can substitute these values into the formula for the slope \( b_1 \):

\[
b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]

Let's compute the slope \( b_1 \).The estimated slope \( b_1 \) of the regression line is \( 0.213 \), rounded to three decimal places.

Would you like more details on this calculation or further clarification?

Here are five related questions you might explore:
1. How is the intercept \( b_0 \) of the regression line calculated?
2. What is the meaning of the slope in this context?
3. How do we determine if the correlation coefficient is statistically significant?
4. What are the assumptions behind linear regression?
5. How can we check if the regression model is a good fit?

**Tip:** Always check the statistical significance of the correlation coefficient before making predictions using a regression model.

The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Price in Dollars: 28, 29, 35, 36, 50. Number of Bids: 2, 3, 4, 6, 7. Find the estimated slope. Round your answer to three decimal places.

Learn how to estimate the slope of a linear regression line using a real-world example involving online auctions. This problem includes calculating the slope using the formula b1 = (n Σxy - Σx Σy) / (n Σx^2 - (Σx)^2) and predicts the number of bids based on item price. Data: Price (28, 29, 35, 36, 50) and Bids (2, 3, 4, 6, 7). Suitable for students in Grades 10-12.

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