The price P per unit at which a company can sell all that it produces is given by the, function P(x) = 300 — 4x. The cost function is c(x) = 500 + 28x where x is the number, of units produced. Find x so that the profit is maximum.,
Question:, 1) Find the value of x.
2) In using regression analysis for making predictions what are the assumptions, involved.,
3) What is a simple linear regression model?,
4) What is a scatter diagram method?
To find the value of x that maximizes profit, we need to calculate the profit function and then find the critical points of that function. The profit function is defined as the difference between the revenue (price per unit multiplied by the number of units sold) and the cost.
1) To find the value of x:
Profit = Revenue - Cost
Profit(x) = P(x) * x - c(x)
Profit(x) = (300 - 4x) * x - (500 + 28x)
Profit(x) = 300x - 4x^2 - 500 - 28x
To find the value of x that maximizes profit, we will need to find the critical points by taking the derivative of the profit function and setting it equal to 0. We can then solve for x.
Profit'(x) = 300 - 8x - 28
Setting Profit'(x) = 0:
300 - 8x - 28 = 0
-8x = -272
x = 34
Therefore, the value of x that maximizes profit is 34.
2) In using regression analysis for making predictions, there are several assumptions involved. These assumptions include:
- Linearity: The relationship between the variables is assumed to be linear.
- Independence: The observations or data points are assumed to be independent of each other.
- Homoscedasticity: The variability of the residuals (the differences between the observed and predicted values) should be constant across all levels of the independent variable.
- Normality of residuals: The residuals should follow a normal distribution.
- No multicollinearity: The independent variables should not be highly correlated with each other.
3) A simple linear regression model is a statistical method used to determine the relationship between two variables - one independent variable and one dependent variable. The dependent variable is assumed to be a linear function of the independent variable. The model can be represented by the equation Y = a + bX, where Y is the dependent variable, X is the independent variable, a is the intercept, and b is the slope.
4) A scatter diagram method, also known as a scatter plot, is a graphical representation of the relationship between two variables. It is typically used in exploratory data analysis to visualize the pattern or trend between the variables. The data points are plotted on a graph with one variable on the x-axis and the other variable on the y-axis. By examining the scatter plot, we can determine if there is a positive, negative, or no correlation between the variables.