Mathematics, 11.06.2020 03:57, srice6

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 Dollars2334404647Number of Bids13457Step 1 of 6:Find the estimated slope. Round your answer to three decimal places. Step 2 of 6:Find the estimated y-intercept. Round your answer to three decimal places. Step 3 of 6:Determine if the statement "All points predicted by the linear model fall on the same line" is true or false. Step 4 of 6:Find the estimated value of y when x=46. Round your answer to three decimal places. Step 5 of 6:Determine the value of the dependent variable y^ at x=0.Step 6 of 6:Find the value of the coefficient of determination. Round your answer to three decimal places.

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answered: lexus830

1) Estimated slope = b₁ = 0.215

2) Estimated y-intercept = b₀ = -4.185

3) Not all the points predicted fall on the same straight line, but the model gives a close to ideal estimate of the line of best fit.

4) The estimated value of y when x=46 is 5.705

5) The value of the dependent variable y^ at x=0 is -4.185

6) The coefficient of determination = 0.951

Step-by-step explanation:

To solve this, we apply regression analysis

y = b₀ + b₁x

Price in Dollars | 23 | 34 | 40 | 46 | 47

Number of Bids | 1 | 3 | 4 | 5 | 7

For this question, we want to predict the number of bids (dependent variable, y), given the list price of the item (independent variable, x)

So, running the analysis on a spreadsheet application, like excel, the table of parameters is obtained and presented in the first attached image to this solution.

Σxᵢ = sum of all the independent variables (sum of all the list prices)

Σyᵢ = sum of all the dependent variables (sum of all the number of bids in the table)

Σxᵢyᵢ = sum of the product of each dependent variable and its corresponding independent variable

Σxᵢ² = sum of the square of each independent variable (list prices)

Σyᵢ² = sum of the square of each dependent variable (number of bids)

n = number of variables = 5

The scatter plot and the line of best fit is presented in the second attached image to this solution

Then the regression analysis is then done

Slope; m = b₁ = [n×Σxᵢyᵢ - (Σxᵢ)×(Σyᵢ)] / [nΣxᵢ² - (∑xi)²]

Intercept b: b₀ = [Σyᵢ - m×(Σxᵢ)] / n

Mean of x = (Σxᵢ)/n

Mean of y = (Σyᵢ) / n

Sample correlation coefficient r: r =

[n*Σxᵢyᵢ - (Σxᵢ)(Σyᵢ)] ÷ {√([n*Σxᵢ² - (Σxᵢ)²][n*Σyᵢ² - (Σyᵢ)²])}

And -1 ≤ r ≤ +1

All of these formulas are properly presented in the third attached image to this answer

The table of results; mean of x, mean of y, intercept, slope, regression equation and sample coefficient is presented in the fourth attached image to this answer.

Hence, the regression equation is

y = -4.185 + 0.215x

y = b₀ + b₁x

Intercept = b₀ = -4.185

Slope = b₁ = 0.215

And the regression coefficient = 0.951 (Which is very close to 1 and indicates statistic significance)

Hence, we can use this answer obtained to answer the questions attached

1) Find the estimated slope.

Estimated slope = b₁ = 0.215

2) Find the estimated y-intercept.

Estimated y-intercept = b₀ = -4.185

3) Determine if the statement "All points predicted by the linear model fall on the same line" is true or false.

Taking a few of sample data

x = 23 when y = 1

y = -4.185 + 0.215x

y = -4.185 + 0.215 (23) = 0.76 ≈ 1

x = 34, y = 3

y = -4.185 + 0.215 (34) = 3.125 ≈ 3

Hence, it is evident that not all the points predicted fall on the same straight line, but the model gives a close to ideal estimate of the line of best fit.

4) Find the estimated value of y when x=46.

The linear model is

y = -4.185 + 0.215x

when x = 46

y = -4.185 + 0.215(46) = 5.705

5) Determine the value of the dependent variable y^ at x=0.

y = -4.185 + 0.215x

when x = 0

y = -4.185 + 0.215(0) = -4.185

6) Find the value of the coefficient of determination.

The coefficient of determination = regression coefficient = 0.951 (as calculated above)

Hope this Helps!!!