James Profit wants to take National Widget Company public. He is interested in the relationship between the size of the initial public offering and the price per share. A sample of 10 companies that recently went public revealed the following information:
size($M)price per share
9.0 10.0
13.0 12.4
11.0 10.5
14.0 12.8
8.0 13.6
9.0 11.5
10.0 14.2
12.0 9.7
10.0 12.3
7.0 9.6
a. The regression equation is:
b. The coefficient of correlation is:
c. The coefficient of determination is:
d. What would Y equal if X equals 13?
To find the answers to these questions, we need to perform a regression analysis using the given data. This analysis will help us determine the relationship between the size of the initial public offering (X) and the price per share (Y).
a. To find the regression equation, we will use the formula:
Y = a + bX
Where Y represents the dependent variable (price per share), X represents the independent variable (size of the initial public offering), a represents the intercept, and b represents the slope of the line.
The regression equation can be found by calculating the values of a and b using statistical software or spreadsheet program like Excel. These values are determined through a process called linear regression.
b. The coefficient of correlation measures the strength and direction of the linear relationship between the variables X and Y. It ranges between -1 and +1, where +1 indicates a perfect positive correlation, 0 indicates no correlation, and -1 indicates a perfect negative correlation.
To calculate the coefficient of correlation, we will use the following formula:
r = ∑ ((X - X̄)(Y - Ȳ)) / (√(∑(X - X̄)²) * √(∑(Y - Ȳ)²))
Where X̄ represents the mean of X, Ȳ represents the mean of Y, and ∑ denotes the sum of.
c. The coefficient of determination measures the proportion of the total variation in Y that can be explained by the variation in X. It ranges between 0 and 1, where 1 indicates that all the variability in Y can be explained by X.
To calculate the coefficient of determination, we will square the value of the coefficient of correlation (r).
d. To find the value of Y when X equals 13, we will substitute this value into the regression equation we found in part (a). This will give us the predicted price per share (Y) for the given value of X.
Let's perform the necessary calculations using the given data:
size($M) price per share
9.0 10.0
13.0 12.4
11.0 10.5
14.0 12.8
8.0 13.6
9.0 11.5
10.0 14.2
12.0 9.7
10.0 12.3
7.0 9.6
By calculating the regression analysis and applying the formulas mentioned above, we will be able to find the answers to the questions.