CASE STUDY : 2
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?
CASE STUDY : 3
Mr Sehwag invests Rs 2000 every year with a company, which pays interest at 10% p.a.
He allows his deposit to accumulate at C.I. Find the amount to the credit of the person
at the end of 5th year.
Question :
1) What is the Time Value of Money concept.
2) What do you mean by present value of money?
3) What is the Future Value of money.
4) What the amount to be credited at the end of 5th year.
CASE STUDY : 4
The cost of fuel in running of an engine is proportional to the square of the speed and is
Rs 48 per hour for speed of 16 kilometers per hour. Other expenses amount to Rs 300
per hour. What is the most economical speed?
Question:
1) What is most economical speed?
2) What is a chi-square test?
3) What is sampling and what are its uses.
4) Is there any alternative formula to find the value of Chi-square?
CASE STUDY : 2
1) To find the value of x that maximizes the profit, we need to determine the profit function. The profit function can be calculated by subtracting the cost function from the revenue function. In this case, the revenue function is given by P(x) = 300 - 4x, and the cost function is given by c(x) = 500 + 28x. So the profit function, denoted as M(x), can be calculated as follows:
M(x) = P(x) - c(x) = (300 - 4x) - (500 + 28x) = -32x - 200
To find the value of x that maximizes the profit, we need to find the maximum of the function M(x). Since this function is linear, the maximum occurs at the endpoint of its domain. However, since x can represent the number of units produced, it cannot be negative. Therefore, we need to determine the domain of x for which it is reasonable to produce units.
Assuming the company cannot produce negative units and the value of x must be an integer, we can set the inequality:
x ≥ 0
Thus, the value of x that maximizes the profit is x = 0, as it is the only feasible value for x.
2) Regression analysis makes certain assumptions. The basic assumptions include:
- Linearity: The relationship between the dependent variable and the independent variable(s) is linear.
- Independence: The observations in the data set are independent of each other.
- Homoscedasticity: The variance of the errors is constant across all levels of the independent variables.
- Normality: The errors follow a normal distribution.
Violations of these assumptions can affect the accuracy and reliability of the regression analysis results.
3) A simple linear regression model is a statistical model that examines the linear relationship between a dependent variable and one independent variable. It is called "simple" because it involves only one predictor variable. The model assumes that the relationship between the dependent variable and the independent variable(s) can be represented by a straight line.
The simple linear regression model can be expressed using the equation:
Y = β0 + β1*X + ε
Where:
Y is the dependent variable,
X is the independent variable,
β0 and β1 are the regression coefficients (intercept and slope, respectively), and
ε is the error term.
The goal of simple linear regression is to estimate the values of β0 and β1 in order to predict the value of Y for a given X.
4) A scatter diagram, also known as a scatter plot, is a graph that visually displays the relationship between two variables. It is a useful tool for understanding and analyzing the correlation or association between the variables.
To create a scatter diagram, each pair of data points (x, y) is plotted on a graph, where x represents the independent variable, and y represents the dependent variable. The resulting points are then connected to form a scatter plot. The pattern formed by the points on the scatter diagram can provide insights into the nature and strength of the relationship between the variables.