Management Consulting/quantitative methods
case study 1 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 2 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 3 The cost of fuel is running of an engine is proportional to the square of the speed and is Rs48 per hour for speed of 16 kilometers per hour.Other expenses amount to Rs300 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 : 1
1) Find the value of x.
2) In using regression analysis for making predictions what are the assumptions involved.
these assumptions are:
1. Validity. Most importantly, the data you are analyzing should map to the research question you are trying to answer. This sounds obvious but is often overlooked or ignored because it can be inconvenient. . . .
2. Additivity and linearity. The most important mathematical assumption of the regression model is that its deterministic component is a linear function of the separate predictors . . .
3. Independence of errors. . . .
4. Equal variance of errors. . . .
5. Normality of errors. . . .
Further assumptions are necessary if a regression coefficient is to be given a causal interpretation . . .
Normality and equal variance are typically minor concerns, unless you’re using the model to make predictions for individual data points.
The six “steps” to interpreting the result of a regression analysis are:
Look at the prediction equation to see an estimate of the relationship.
Refer to the standard error of the prediction (in the appropriate model) when making predictions for individuals, and the standard error of the estimated mean when estimating the average value of the dependent variable across a large pool of similar individuals.
Refer to the standard errors of the coefficients (in the most complete model) to see how much you can trust the estimates of the effects of the explanatory variables.
Look at the significance levels of the t-ratios to see how strong is the evidence in support of including each of the explanatory variables in the model.
Use the “adjectived” coefficient of determination to measure the potential explanatory power of the model.
Compare the beta-weights of the explanatory variables in order to rank them in order of explanatory importance
3) What is a simple linear regression model?
Simple Linear Regression
Simple linear regression is when you want to predict values of one variable, given values of another variable. For example, you might want to predict a person's height (in inches) from his weight (in pounds). Imagine a sample of ten people for whom you know their height and weight. You could plot the values on a graph, with weight on the x axis and height on the y axis. If there were a perfect linear relationship between height and weight, then all 10 points on the graph would fit on a straight line. But, this is never the case (unless your data are rigged). If there is a (nonperfect) linear relationship between height and weight (presumably a positive one), then you would get a cluster of points on the graph which slopes upward. In other words, people who weigh a lot should be taller than those people who are of less weight. (See graph below.)
The purpose of regression analysis is to come up with an equation of a line that fits through that cluster of points with the minimal amount of deviations from the line. The deviation of the points from the line is called "error." Once you have this regression equation, if you knew a person's weight, you could then predict their height. Simple linear regression is actually the same as a bivariate correlation between the independent and dependent variable.
Standard Multiple Regression
Standard multiple regression is the same idea as simple linear regression, except now you have several independent variables predicting the dependent variable. To continue with the previous example, imagine that you now wanted to predict a person's height from the gender of the person and from the weight. You would use standard multiple regression in which gender and weight were the independent variables and height was the dependent variable. The resulting output would tell you a number of things. First, it would tell you how much of the variance of height was accounted for by the joint predictive power of knowing a person's weight and gender. This value is denoted by "R2". The output would also tell you if the model allows you to predict a person's height at a rate better than chance. This is denoted by the significance level of the overall F of the model. If the significance is .05 (or less), then the model is considered significant. In other words, there is only a 5 in a 100 chance (or less) that there really is not a relationship between height and weight and gender. For whatever reason, within the social sciences, a significance level of .05 is often considered the standard for what is acceptable. If the significance level is between .05 and .10, then the model is considered marginal. In other words, the model is fairly good at predicting a person's height, but there is between a 5-10% probability that there really is not a relationship between height and weight and gender.
In addition to telling you the predictive value of the overall model, standard multiple regression tells you how well each independent variable predicts the dependent variable, controlling for each of the other independent variables. In our example, then, the regression would tell you how well weight predicted a person's height, controlling for gender, as well as how well gender predicted a person's height, controlling for weight.
To see if weight was a "significant" predictor of height you would look at the significance level associated with weight on the printout. Again, significance levels of .05 or lower would be considered significant, and significance levels .05 and .10 would be considered marginal. Once you have determined that weight was a significant predictor of height, then you would want to more closely examine the relationship between the two variables. In other words, is the relationship positive or negative? In this example, we would expect that there would be a positive relationship. In other words, we would expect that the greater a person's weight, the greater his height. (A negative relationship would be denoted by the case in which the greater a person's weight, the shorter his height.) We can determine the direction of the relationship between weight and height by looking at the regression coefficient associated with weight. There are two kinds of regression coefficients: B (unstandardized) and beta (standardized). The B weight associated with each variable is given in terms of the units of this variable. For weight, the unit would be pounds, and for height, the unit is inches. The beta uses a standard unit that is the same for all variables in the equation. In our example, this would be a unit of measurement that would be common to weight and height. Beta weights are useful because then you can compare two variables that are measured in different units, as are height and weight.
If the regression coefficient is positive, then there is a positive relationship between height and weight. If this value is negative, then there is a negative relationship between height and weight. We can more specifically determine the relationship between height and weight by looking at the beta coefficient for weight. If the beta = .35, for example, then that would mean that for one unit increase in weight, height would increase by .35 units. If the beta=-.25, then for one unit increase in weight, height would decrease by .25 units. Of course, this relationship is valid only when holding gender constant.
A similar procedure would be done to see how well gender predicted height. However, because gender is a dichotomous variable, the interpretation of the printouts is slightly different. As with weight, you would check to see if gender was a significant predictor of height, controlling for weight. The difference comes when determining the exact nature of the relationship between gender and height. That is, it does not make sense to talk about the effect on height as gender increases or decreases (sex is not measured as a continuous variable). Imagine that gender had been coded as either 0 or 1, with 0 = female and 1=male. If the beta coefficient of gender were positive, this would mean that males are taller than females. If the beta coefficient of gender were negative, this would mean that males are shorter than females. Looking at the magnitude of the beta, you can more closely determine the relationship between height and gender. Imagine that the beta of gender were .25. That means that males would be .25 units taller than females. Conversely, if the beta coefficient were -.25, this would mean that males were .25 units shorter than females. Of course, this relationship would be true only when controlling for weight.
As mentioned, the significance levels given for each independent variable indicates whether that particular independent variable is a significant predictor of the dependent variable, over and above the other independent variables. Because of this, an independent variable that is a significant predictor of a dependent variable in simple linear regression may not be significant in multiple regression (i.e., when other independent variables are added into the equation). This could happen because the variance that the first independent variable shares with the dependent variable could overlap with the variance that is shared between the second independent variable and the dependent variable. Consequently, the first independent variable is no longer uniquely predictive and thus would not show up as being significant in the multiple regression. Because of this, it is possible to get a highly significant R2, but have none of the
the independent variables be significant.
4) What is a scatter diagram method?
A visual technique for separating the fixed and variable elements of a semi-variable expense (also called a mixed expense) in order to estimate and budget for future costs. A scattergraph is made up of a horizontal x axis that represents production activity, a vertical y axis that represents cost, data that are plotted as points on the graph and a regression line that runs through the dots which represents the relationship between the variables.
Business managers use the scattergraph method in cost estimation to anticipate operating costs at different activity levels. The method gets its name from the overall image of the graph, which consists of many scattered dots. While the method is simple, it is also imprecise. Alternate methods of cost estimation include the high-low method, account analysis and least squares.
What it is:
A scatter diagram is a tool for analyzing relationships between two variables. One variable is
plotted on the horizontal axis and the other is plotted on the vertical axis. The pattern of their
intersecting points can graphically show relationship patterns. Most often a scatter diagram is used
to prove or disprove cause-and-effect relationships. While the diagram shows relationships, it
does not by itself prove that one variable causes the other. In addition to showing possible causeand-
effect relationships, a scatter diagram can show that two variables are from a common cause
that is unknown or that one variable can be used as a surrogate for the other.
When to use it:
Use a scatter diagram to examine theories about cause-and-effect relationships and to search for
root causes of an identified problem.
Use a scatter diagram to design a control system to ensure that gains from quality improvement
efforts are maintained.
How to use it:
Collect data. Gather 50 to 100 paired samples of data that show a possible relationship.
Draw the diagram. Draw roughly equal horizontal and vertical axes of the diagram, creating a
square plotting area. Label the axes in convenient multiples (1, 2, 5, etc.) increasing on the
horizontal axes from left to right and on the vertical axis from bottom to top. Label both axes.
Plot the paired data. Plot the data on the chart, using concentric circles to indicate repeated data
Title and label the diagram.
Interpret the data. Scatter diagrams will generally show one of six possible correlations
between the variables:
Strong Positive Correlation The value of Y clearly increases as the value of X increases.
Strong Negative Correlation The value of Y clearly decreases as the value of X increases.
Weak Positive Correlation The value of Y increases slightly as the value of X increases.
Weak Negative Correlation The value of Y decreases slightly as the value of X increases.
Complex Correlation The value of Y seems to be related to the value of X, but the
relationship is not easily determined.
No Correlation There is no demonstrated connection between the two variables.
Scatter Diagram Example
Strong Positive Correlation
0 2 4 6
Strong Negative Correlation
0 2 4 6
Weak Positive Correlation
0 2 4 6
Weak Negative Correlation
0 2 4 6
0 2 4 6
0 2 4 6
Scatter Graph Method
Scatter graph method is a graphical technique of separating fixed and variable components of mixed cost by plotting activity level along x-axis and corresponding total cost (mixed cost) along y-axis. A regression line is then drawn on the graph by visual inspection. The line thus drawn is used to estimate the total fixed cost and variable cost per unit. The point where the line intercepts y-axis is the estimated fixed cost and the slope of the line is the average variable cost per unit. Since the visual inspection does not involve any mathematical testing therefore this method should be applied with great care.
Step 1: Draw scatter graph
Plot the data on scatter graph. Plot activity level (i.e. number of units, labor hours etc.) along x-axis and total mixed cost along y-axis.
Step 2: Draw regression line
Draw a regression line over the scatter graph by visual inspection and try to minimize the total vertical distance between the line and all the points. Extend the line towards y-axis.
Step 3: Find total fixed cost
Total fixed is given by the y-intercept of the line. Y-intercept is the point at which the line cuts y-axis.
Step 4: Find variable cost per unit
Variable cost per unit is equal to the slope of the line. Take two points (x1,y1) and (x2,y2) on the line and calculate variable cost using the following formula:
Variable Cost per Unit = Slope of Regression Line = y2 − y1
x2 − x1
Company α decides to use scatter graph method to split its factory overhead (FOH) into variable and fixed components. Following is the data which is provided for the analysis.
Month Units FOH
1 1,520 $36,375
2 1,250 38,000
3 1,750 41,750
4 1,600 42,360
5 2,350 55,080
6 2,100 48,100
7 3,000 59,000
8 2,750 56,800
Fixed Cost = y-intercept = $18,000
Variable Cost per Unit = Slope of Regression Line
To calculate slop we will take two points on line: (0,18000) and (3500,68000)
Variable Cost per Unit = (68000 − 18000) ÷ (3500 − 0) = $14.286
CASE STUDY : 2
What is the Time Value of Money concept.
'Time Value of Money - TVM'
The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received.
Everyone knows that money deposited in a savings account will earn interest. Because of this universal fact, we would prefer to receive money today rather than the same amount in the future.
For example, assuming a 5% interest rate, $100 invested today will be worth $105 in one year ($100 multiplied by 1.05). Conversely, $100 received one year from now is only worth $95.24 today ($100 divided by 1.05), assuming a 5% interest rate.
What Is Time Value?
If you're like most people, you would choose to receive the $10,000 now. After all, three years is a long time to wait. Why would any rational person defer payment into the future when he or she could have the same amount of money now? For most of us, taking the money in the present is just plain instinctive. So at the most basic level, the time value of money demonstrates that, all things being equal, it is better to have money now rather than later.
But why is this? A $100 bill has the same value as a $100 bill one year from now, doesn't it? Actually, although the bill is the same, you can do much more with the money if you have it now because over time you can earn more interest on your money.
Back to our example: by receiving $10,000 today, you are poised to increase the future value of your money by investing and gaining interest over a period of time. For Option B, you don't have time on your side, and the payment received in three years would be your future value. To illustrate, we have provided a timeline:
If you are choosing Option A, your future value will be $10,000 plus any interest acquired over the three years. The future value for Option B, on the other hand, would only be $10,000. So how can you calculate exactly how much more Option A is worth, compared to Option B? Let's take a look.
Future Value Basics
If you choose Option A and invest the total amount at a simple annual rate of 4.5%, the future value of your investment at the end of the first year is $10,450, which of course is calculated by multiplying the principal amount of $10,000 by the interest rate of 4.5% and then adding the interest gained to the principal amount:
Future value of investment at end of first year:
= ($10,000 x 0.045) + $10,000
You can also calculate the total amount of a one-year investment with a simple manipulation of the above equation:
Original equation: ($10,000 x 0.045) + $10,000 = $10,450
Manipulation: $10,000 x [(1 x 0.045) + 1] = $10,450
Final equation: $10,000 x (0.045 + 1) = $10,450
The manipulated equation above is simply a removal of the like-variable $10,000 (the principal amount) by dividing the entire original equation by $10,000.
If the $10,450 left in your investment account at the end of the first year is left untouched and you invested it at 4.5% for another year, how much would you have? To calculate this, you would take the $10,450 and multiply it again by 1.045 (0.045 +1). At the end of two years, you would have $10,920:
Future value of investment at end of second year:
= $10,450 x (1+0.045)
The above calculation, then, is equivalent to the following equation:
Future Value = $10,000 x (1+0.045) x (1+0.045)
Think back to math class and the rule of exponents, which states that the multiplication of like terms is equivalent to adding their exponents. In the above equation, the two like terms are (1+0.045), and the exponent on each is equal to 1. Therefore, the equation can be represented as the following:
We can see that the exponent is equal to the number of years for which the money is earning interest in an investment. So, the equation for calculating the three-year future value of the investment would look like this:
This calculation shows us that we don't need to calculate the future value after the first year, then the second year, then the third year, and so on. If you know how many years you would like to hold a present amount of money in an investment, the future value of that amount is calculated by the following equation:
Present Value Basics
If you received $10,000 today, the present value would of course be $10,000 because present value is what your investment gives you now if you were to spend it today. If $10,000 were to be received in a year, the present value of the amount would not be $10,000 because you do not have it in your hand now, in the present. To find the present value of the $10,000 you will receive in the future, you need to pretend that the $10,000 is the total future value of an amount that you invested today. In other words, to find the present value of the future $10,000, we need to find out how much we would have to invest today in order to receive that $10,000 in the future.
To calculate present value, or the amount that we would have to invest today, you must subtract the (hypothetical) accumulated interest from the $10,000. To achieve this, we can discount the future payment amount ($10,000) by the interest rate for the period. In essence, all you are doing is rearranging the future value equation above so that you may solve for P. The above future value equation can be rewritten by replacing the P variable with present value (PV) and manipulated as follows:
Let's walk backwards from the $10,000 offered in Option B. Remember, the $10,000 to be received in three years is really the same as the future value of an investment. If today we were at the two-year mark, we would discount the payment back one year. At the two-year mark, the present value of the $10,000 to be received in one year is represented as the following:
Present value of future payment of $10,000 at end of year two:
Note that if today we were at the one-year mark, the above $9,569.38 would be considered the future value of our investment one year from now.
Continuing on, at the end of the first year we would be expecting to receive the payment of $10,000 in two years. At an interest rate of 4.5%, the calculation for the present value of a $10,000 payment expected in two years would be the following:
Present value of $10,000 in one year:
Of course, because of the rule of exponents, we don't have to calculate the future value of the investment every year counting back from the $10,000 investment at the third year. We could put the equation more concisely and use the $10,000 as FV. So, here is how you can calculate today's present value of the $10,000 expected from a three-year investment earning 4.5%:
So the present value of a future payment of $10,000 is worth $8,762.97 today if interest rates are 4.5% per year. In other words, choosing Option B is like taking $8,762.97 now and then investing it for three years. The equations above illustrate that Option A is better not only because it offers you money right now but because it offers you $1,237.03 ($10,000 - $8,762.97) more in cash! Furthermore, if you invest the $10,000 that you receive from Option A, your choice gives you a future value that is $1,411.66 ($11,411.66 - $10,000) greater than the future value of Option B.
Present Value of a Future Payment
Let's add a little spice to our investment knowledge. What if the payment in three years is more than the amount you'd receive today? Say you could receive either $15,000 today or $18,000 in four years. Which would you choose? The decision is now more difficult. If you choose to receive $15,000 today and invest the entire amount, you may actually end up with an amount of cash in four years that is less than $18,000. You could find the future value of $15,000, but since we are always living in the present, let's find the present value of $18,000 if interest rates are currently 4%. Remember that the equation for present value is the following:
In the equation above, all we are doing is discounting the future value of an investment. Using the numbers above, the present value of an $18,000 payment in four years would be calculated as the following:
From the above calculation we now know our choice is between receiving $15,000 or $15,386.48 today. Of course we should choose to postpone payment for four years!
The Bottom Line
These calculations demonstrate that time literally is money - the value of the money you have now is not the same as it will be in the future and vice versa. So, it is important to know how to calculate the time value of money so that you can distinguish between the worth of investments that offer you returns at different times.
What do you mean by present value of money?
'Present Value - PV'
The current worth of a future sum of money or stream of cash flows given a specified rate of return. Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they be earnings or obligations.
This sounds a bit confusing, but it really isn't. The basis is that receiving $1,000 now is worth more than $1,000 five years from now, because if you got the money now, you could invest it and receive an additional return over the five years.
The calculation of discounted or present value is extremely important in many financial calculations. For example, net present value, bond yields, spot rates, and pension obligations all rely on the principle of discounted or present value. Learning how to use a financial calculator to make present value calculations can help you decide whether you should accept a cash rebate, 0% financing on the purchase of a car or to pay points on a mortgage.
Present value describes how much a future sum of money is worth today.
How it works/Example:
The formula for present value is:
PV = CF/(1+r)n
CF = cash flow in future period
r = the periodic rate of return or interest (also called the discount rate or the required rate of return)
n = number of periods
Let's look at an example. Assume that you would like to put money in an account today to make sure your child has enough money in 10 years to buy a car. If you would like to give your child $10,000 in 10 years, and you know you can get 5% interest per year from a savings account during that time, how much should you put in the account now? The present value formula tells us:
PV = $10,000/ (1 + .05)10 = $6,139.13
Thus, $6,139.13 will be worth $10,000 in 10 years if you can earn 5% each year. In other words, the present value of $10,000 in this scenario is $6,139.13.
It is important to note that the three most influential components of present value are time, expected rate of return, and the size of the future cash flow. To account for inflation in the calculation, investors should use the real interest rate (nominal interest rate - inflation rate). If given enough time, small changes in these components can have significant effects.
Why it Matters:
The concept of present value is one of the most fundamental and pervasive in the world of finance. It is the basis for stock pricing, bond pricing, financial modeling, banking, insurance, pension fund valuation, and even lottery payouts. It accounts for the fact that money we receive today can be invested today to earn a return. In other words, present value accounts for the time value of money.
In the stock world, calculating present value can be a complex, inexact process that incorporates assumptions regarding short and long-term growth rates, capital expenditures, return requirements, and many other factors. Naturally, such variables are impossible to predict with perfect precision. Regardless, present value provides an estimate of what we should spend today (e.g., what price we should pay) to have an investment worth a certain amount of money at a specific point in the future -- this is the basic premise of the math behind most stock- and bond-pricing models.
Present value is one of the most important concepts in finance. Luckily, it's easy to calculate once you know a few tricks..
3) What is the Future Value of money.
'Future Value - FV'
The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. There are two ways to calculate FV:
1) For an asset with simple annual interest: = Original Investment x (1+(interest rate*number of years))
2) For an asset with interest compounded annually: = Original Investment x ((1+interest rate)^number of years)
Consider the following examples:
1) $1000 invested for 5 years with simple annual interest of 10% would have a future value of $1,500.00.
2) $1000 invested for 5 years at 10%, compounded annually has a future value of $1,610.51.
Future value (FV) refers to a method of calculating how much the present value (PV) of an asset or cash will be worth at a specific time in the future.
How it works/Example:
There are two ways of calculating future value: simple annual interest and annual compound interest.
Future value with simple interest is calculated in the following manner:
Future Value = Present Value x [1 + (Interest Rate x Number of Years)]
For example, Bob invests $1,000 for five years with an interest rate of 10%. The future value would be $1,500.
Future Value = $1,000 x [1 + (0.1 x 5)]
Future Value = $1,000 x 1.5
Future Value = $1,500
Future value with compounded interest is calculated in the following manner:
Future Value = Present Value x [(1 + Interest Rate) Number of Years]
For example, John invests $1,000 for five years with an interest rate of 10%, compounded annually. The future value of John's investment would be $1,610.51.
Future Value = $1,000 x [(1 + 0.1)5]
Future Value = $1,000 x 1.61051
Future Value = $1,610.51
It is important to remember that simple interest is always based on the present value, whereas compounded interest means that the present value grows exponentially each year.
Why it Matters:
Although calculating future value has its benefits, it is important to remember that future value does not include adjustments for inflation, fluctuating interest rates or fluctuating currency values that are likely to affect the true value of money or assets in the future.
CASE STUDY : 3
What is most economical speed?
Economic speed is the optimal rate of speed that a vehicle can go to obtain the best financial results. This depends on such factors as the price of gas, the gas mileage, and the engine power of the vehicle. Other considerations include the conditions of the road and how busy or congested the road is at the time of driving. There are calculations that can be done to determine the economic speed in different situations and with various vehicles. When an individual is trying to get the most money out of his or her car, finding out the economic speed is a helpful tool.
When a vehicle is going fast, its overall economic speed is generally reduced because it uses more gas in the process. This means that it is costing more money to drive the vehicle than it would if it were going at a slower rate due to the cost of gas. Fast driving also puts more wear and tear on a vehicle than slower driving does. If a vehicle goes too slow, however, the economic speed is reduced as well because it is costing time to get to a destination that could have been used to make money.
Bad road conditions contribute to a lower economic speed because they cause the driver to use more gas to navigate them, as well as placing more wear and tear on the vehicle. This is difficult to calculate, however, because it is hard to determine the exact conditions of a road over a period of time. Traffic also reduces economic speed because when a vehicle is constantly braking and accelerating, additional gas is used in the process of driving.
Basic economic speed can be calculated by filling a vehicle with gas and recording the number of miles that the vehicle has on it. At this point, the driver can drive to a destination going the speed that he or she wishes to test. Once the vehicle's tank is half-empty, it should be refilled and the mileage recorded once again. The first mileage can then be subtracted from the second mileage and divided by the amount of gas, in gallons, that it took to fill the car up the second time. This is the miles per gallon (MPG) for driving that speed, so the test can be performed again at a different speed to see which is the best economic speed for the vehicle.
What speed should I drive to get maximum fuel efficiency?
Martyn Goddard/Getty Images
In general, smaller, lighter, more aerodynamic cars will get their best mileage at higher speeds. Bigger, heavier, less aerodynamic vehicles will get their best mileage at lower speeds. See sports car pictures.
This is actually a pretty complicated question. What you are asking is what constant speed will give the best mileage. We won't talk about stops and starts. We'll assume you are going on a very long highway trip and want to know what speed will give you the best mileage. We'll start by discussing how much power it takes to push the car down the road.
The power to push a car down the road varies with the speed the car is traveling. The power required follows an equation of the following form:
road load power = av + bv² + cv³
The letter v represents the velocity of the car, and the letters a, b and c represent three different constants:
The a component comes mostly from the rolling resistance of the tires, and friction in the car's components, like drag from the brake pads, or friction in the wheel bearings.
The b component also comes from friction in components, and from the rolling resistance in the tires. But it also comes from the power used by the various pumps in the car.
The c component comes mostly from things that affect aerodynamic drag like the frontal area, drag coefficient and density of the air.
These constants will be different for every car. But the bottom line is, if you double your speed, this equation says that you will increase the power required by much more than double. A hypothetical medium sized SUV that requires 20 horsepower at 50 mph might require 100 horsepower at 100 mph.
You can also see from the equation that if the velocity v is 0, the power required is also 0. If the velocity is very small then the power required is also very small. So you might be thinking that you would get the best mileage at a really slow speed like 1 mph.
But there is something going on in the engine that eliminates this theory. If your car is going 0 mph your engine is still running. Just to keep the cylinders moving and the various fans, pumps and generators running consumes a certain amount of fuel. And depending on how many accessories (such as headlights and air conditioning) you have running, your car will use even more fuel.
So even when the car is sitting still it uses quite a lot of fuel. Cars get the very worst mileage at 0 mph; they use gasoline but don't cover any miles. When you put the car in drive and start moving at say 1 mph, the car uses only a tiny bit more fuel, because the road load is very small at 1 mph. At this speed the car uses about the same amount of fuel, but it went 1 mile in an hour. This represents a dramatic increase in mileage. Now if the car goes 2 mph, again it uses only a tiny bit more fuel, but goes twice as far. The mileage almost doubled!
Efficiency of an Engine
In effect the efficiency of the engine is improving. It uses a fixed amount of fuel to power itself and the accessories, and a variable amount of fuel depending on the power required to keep the car going at a given speed. So in terms of fuel used per mile, the faster the car goes, the better use we make of that fixed amount of fuel required.
This trend continues to a point. Eventually, that road load curve catches up with us. Once the speed gets up into the 40 mph range each 1 mph increase in speed represents a significant increase in power required. Eventually, the power required increases more than the efficiency of the engine improves. At this point the mileage starts dropping. Let's plug some speeds into our equation and see how a 1 mph increase from 2 to 3 mph compares with a 1 mph increase from 50 to 51 mph. To make things easy we'll assume a, b and c are all equal to 1.
Speed Equation Result
3 mph 3+3²+3³ 39
2 mph 2+2²+2³ 14
Power Increase 25
51 mph 51+51²+51³ 135,303
50 mph 50+50²+50³ 127,550
Power Increase 7,753
You can see that the increase in power required to go from 50 to 51 mph is much greater than to go from 2 to 3 mph.
So, for most cars, the "sweet spot" on the speedometer is in the range of 40-60 mph. Cars with a higher road load will reach the sweet spot at a lower speed. Some of the main factors that determine the road load of the car are:
Coefficient of drag. This is an indicator of how aerodynamic a car is due only to its shape. The most aerodynamic cars today have a drag coefficient that is about half that of some pickups and SUVs.
Frontal area. This depends mostly on the size of the car. Big SUVs have more than double the frontal area of some small cars.
Weight. This affects the amount of drag the tires put on the car. Big SUVs can weigh two to three times what the smallest cars weigh.
In general, smaller, lighter, more aerodynamic cars will get their best mileage at higher speeds. Bigger, heavier, less aerodynamic vehicles will get their best mileage at lower speeds.
If you drive your car in the "sweet spot" you will get the best possible mileage for that car. If you go faster or slower, the mileage will get worse, but the closer you drive to the sweet spot the better mileage you will get.
TIPS FOR FUEL-EFFICIENT DRIVING:
Avoid aggressive driving. "Jack-rabbit" starts and hard braking can increase fuel consumption by as much as 40%. Tests show that "jackrabbit" starts and hard braking reduces travel time by only four percent, while toxic emissions were more than five times higher. The proper way is to accelerate slowly and smoothly, then get into high gear as quickly as possible. In city driving, nearly 50% of the energy needed to power your car goes to acceleration.
Drive steadily at posted speed limits.
Increasing your highway cruising speed from 55mph (90km/h) to 75mph (120km/h) can raise fuel consumption as much as 20%. You can improve your gas mileage 10 - 15% by driving at 55mph rather than 65mph (104km/h).
Note how quickly efficiency drops after 60 mph.
Please Note: The graph above is based on EPA and fueleconomy.gov statistics. There are many factors which affect fuel economy, however, and these figures can vary significantly.
To calculate your own mileage, read: How to calculate gas mileage
Avoid idling your vehicle, in both summer and winter. Idling wastes fuel, gets you nowhere and produces unnecessary greenhouse gases. If you're going to be stopped for more than 30 seconds, except in traffic, turn off the engine. In winter, don't idle a cold engine for more than 30 seconds before driving away. (Older vehicles, however, may need more idling time when first started. In cold, winter conditions all vehicles may need more idling time to warm up and ensure the windshield is fully defogged. Be sure your vehicle is warmed enough to prevent stalling when you pull out.)
Make sure your tires are properly inflated to prevent increased rolling resistance.
Under-inflated tires can cause fuel consumption to increase by as much as 6%.
Check tire pressure at least once a month, when the tires are 'cold' (i.e. when the vehicle has not been driven for at least three hours or for more than 2km). Start by checking tire pressures in your driveway. Note any tire that is underinflated, and then drive to the nearest gas station to add air. Check tire pressures again at the station, and inflate the low tires to the same level as the others (these will likely have higher pressure than they did in the driveway, since the tires have heated up.)
Radial tires can be under inflated yet still look normal. Always use your own tire gauge for consistent results. On average, tires lose about 1 psi per month and 1 psi for every 10 degree drop in temperature.
To determine the correct tire inflation for your car, consult the car's operator manual or ask your tire dealer. Do not inflate your tires to the 'maximum allowed' pressure which is marked on the side of your tires.
According to the Energy Information Administration, tire efficiency could save approximately 800,000 barrels of oil a day.
Select the right gear. Change up through the gears and into top gear as soon as possible without accelerating harder than necessary. Driving in a gear lower than you need wastes fuel; so does letting the engine labour in top gear on hills and corners. Automatic transmissions will shift up more quickly and smoothly if you ease back slightly on the accelerator once the car gathers momentum.
Use your air conditioner sparingly on older cars. Using a vehicle’s air conditioner on a hot summer day can increase fuel consumption as much as 10% in city driving. If it’s cool enough, use the flow-through ventilation on your car instead of the air conditioner. At low speeds, opening the window will also save reduce fuel consumption by reducing A/C use. At higher speeds however, using the A/C may be more efficient than the wind resistance from open windows and sunroof.
Later model cars have more efficient air conditioning units, and the fuel saved by shutting down the A/C is not significant. In newer vehicles, roll up the windows and enjoy the A/C during hot weather.
Use the cruise control. On long stretches of highway driving, cruise control can save fuel by helping your car maintain a steady speed. However, this efficiency is lost on steep hills where the cruise control tries to maintain even speeds. In hilly terrain, it is best to turn off the cruise control.
Choose the octane fuel which best suits your car. Premium, high-octane fuels aren't necessarily the best choice for your car; higher price doesn't guarantee better performance. In fact, such fuels don't provide any greater fuel efficiency. Many cars are designed to use regular low-octane fuel. Check your owner's manual to see what your car requires.
Service your vehicle regularly, according to the manufacturer's instructions. A poorly tuned engine can use up to 50% more fuel and produces up to 50% more emissions than one that is running properly.
Air filters: Dirty air filters can also cause your engine to run at less than peak efficiency Regular visual checks of the air filter will tell you if it needs replacing and your owner's manual will also recommend appropriate replacement intervals. Clogged filters can cause up to a 10% increase in fuel consumption.
Oil: Using the correct viscosity oil is important because higher viscosity oils have greater resistance to the moving parts of the engine, and therefore use more gas. Clean oil also contributes to better gas mileage. It is usually recommended that engine oil be changed every three to five thousand miles.
Monitor power accessories. Be sure to shut off all power-consuming accessories before turning off the ignition. That way, you decrease engine load the next time you start up. Items that plug into your vehicle's cigarette lighter, such as TV consoles for mini-vans and SUVs, can cause the alternator to work harder to provide electrical current. This adds a load to the engine and added load increases fuel use, decreasing your gas mileage.
Tighten your gas cap. If you don't tighten up the gas cap to the second click, gas can evaporate. According to the Car Care Council (carcare.org), loose, missing or damaged gas caps cause 147 million gallons of gas to evaporate every year. Think "aerodynamic" and "lightweight". Reduce drag. Out on the open highway, keep windows rolled up to reduce drag. Remove bicycle and ski racks when not in use. Excess weight also uses more fuel. Remove unnecessary items from inside the vehicle, trunk or truck bed. An extra 100lbs (48 kg) of weight can increase your fuel bill by 2%.
Park in shady areas when possible. Besides helping to keep your car cool, which reduces the need for air conditioning, parking in the shade also minimizes the loss of gas due to evaporation.
Cold weather driving? Use a block heater when the winter temperature drops to -20°C or below. A block heater keeps your engine oil and coolant warm, which makes the vehicle easier to start and can reduce winter fuel consumption by as much as 10%. Use a timer to switch on the block heater one or two hours before you plan to drive.
If you're in the market for a new car, why not purchase the most fuel-efficient model that meets your needs? In Canada, look for the EnerGuide label posted on all new cars, vans and light-duty trucks. The label provides the vehicle's fuel consumption rating and estimated annual fuel costs. If you can't find the EnerGuide label on a vehicle, ask the dealer for its fuel consumption rating.
Plan your trip, whether you are going across town or across the country. Try to combine several errands in one outing, and plan your route to avoid heavy traffic areas, road construction, hilly trerrain, etc. With a little organization, you can group your "town tasks" into fewer trips, saving you time and fuel expense.
Make a commitment to drive less, by walking to some nearby destinations. It's good for your health and the environment. Approximately 50% of car use is for trips within 3 miles of the home. This distance is within the range for easy biking, so it makes sense to try to use your bike for some of these short hops. You'll be saving fuel and reducing pollution, and you can also save on trips to the gym with this added exercise.
Changing the oil in your car?
Disposing of used motor oil by pouring it into storm or sewer drains, dumping it onto the ground, or placing it with household trash may create risks to human health and the environment.
Human health is affected if rainwater carries metal-laden oil into underground streams and contaminates drinking water. It is almost impossible to clean up groundwater once it has been contaminated. Surface runoff from ground disposal and oil poured down drains often lead to water treatment plants, streams or rivers, which can also affect drinking water supplies.
Used oil from a single oil change can ruin a million gallons of fresh water, a year's supply for 50 people.
Pour all collected used oil into a clean, empty, sealable container such as a plastic milk jug. Specialized used oil containers can be purchased at local auto parts stores. Take it to a used oil collection site (UOCS) that accepts and recycles used motor oil. These sites, generally places such as service stations that sell motor oil, can be identified by an amber and black "Recycle Oil" logo.
In the US call 1 800 CLEANUP, and in Canada call 1 800-667-4321 for the nearest used oil disposal facility.
"Hi, Your site says "You can boost the overall fuel-efficiency of your car as much as 30% by ..."
Make that closer to 40%. I get 72 MPG regularly, but my 2003 VW Jetta TDi's official EPA combined estimate is only 45. No custom tech, just efficient driving. And that's not pure highway driving either. My last tank was 69.9 MPG with 40% city driving. The car's efficient diesel engine helps too. Also, the Wall Street Journal mentioned a Honda Insight owner who gets 100 MPG on occasion -- that's fully 58% better than the official combined of 63 MPG". Alexander Passmoore
"Not all cars get the best economy at 55MPH, it depends on gear ratio, I have had cars that deliver better fuel economy at 70 MPH, Like a 1970 Plymouth Duster I had years ago, it got over 21MPG at 70 MPH pulling a double snowmobile trailer with two machines on it and it was over 340 Horse Power. And my 1986 Corvette will get 32 MPH at a steady 70MPH, and only about 28MPH at 55 MPH. Thank You" Richard Heater
What is a chi-square test?
Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel's laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the "goodness to fit" between the observed and expected. Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors. How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected. The chi-square test is always testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result.
The formula for calculating chi-square ( 2) is:
That is, chi-square is the sum of the squared difference between observed (o) and the expected (e) data (or the deviation, d), divided by the expected data in all possible categories.
As a test for independence of attributes
If there are two categorical variables, and our interest is to examine whether these two variables are associated with each other, the chi-square (χ^2 ) test of independence is the correct tool to use. This test is very popular in analyzing cross-tabulations in which an investigator is keen to find out whether the two attributes of interest have any relationship with each other.
The cross-tabulation is popularly called by the term “contingency table”. It contains frequency data that correspond to the categorical variables in the row and column. The marginal totals of the rows and columns are used to calculate the expected frequencies that will be part of the computation of the c² statistic. For calculations on expected frequencies, refer hyper stat on χ^2 test.
Example: A marketing firm producing detergents is interested in studying the consumer behavior in the context of purchase decision of detergents in a specific market. This company is a major player in the detergent market that is characterized by intense competition. It would like to know in particular whether the income level of the consumers influence their choice of the brand. Currently
there are four brands in the market. Brand 1 and Brand 2 are the premium brands while Brand 3 and Brand 4 are the economy brands.
A representative stratified random sampling procedure was adopted covering the entire market using income as the basis of selection. The categories that were used in classifying income level are: Lower, Middle, Upper Middle and High. A sample of 600 consumers participated in this study. The following data emerged from the study.
Cross Tabulation of Income versus Brand chosen (Figures in the cells represent number of consumers)
Brand 1 Brand 2 Brand 3 Brand 4 Total
Lower 25 15 55 65 160
Middle 30 25 35 30 120
Upper Middle 50 55 20 22 147
Upper 60 80 15 18 173
Total 165 175 125 135 600
Analyze the cross-tabulation data above using chi-square test of independence and draw your conclusions.
Null Hypothesis: There is no association between the brand preference and income level (These two attributes are independent).
Alternative Hypothesis: There is association between brand preference and income level (These two attributes are dependent).
Let us take a level of significance of 5%.
In order to calculate the χ^2 value, you need to work out the expected frequency in each cell in the contingency table. In our example, there are 4 rows and 4 columns amounting to 16 elements. There will be 16 expected frequencies. For calculating expected frequencies, please go through hyper stat. Relevant data tables are given below:
Observed Frequencies (These are actual frequencies observed in the survey)
Brand 1 Brand 2 Brand 3 Brand 4 Total
Lower 25 15 55 65 160
Middle 30 25 35 30 120
Upper Middle 50 55 20 22 147
Upper 60 80 15 18 173
Total 165 175 125 135 600
Expected Frequencies (These are calculated on the assumption of the null hypothesis being true: That is, income level and brand preference are independent)
Brand 1 Brand 2 Brand 3 Brand 4 Total
Lower 44.000 46.667 33.333 36.000 160.000
Middle 33.000 35.000 25.000 27.000 120.000
Upper Middle 40.425 42.875 30.625 33.075 147.000
Upper 47.575 50.458 36.042 38.925 173.000
Total 165.000 175.000 125.000 135.000 600.000
There are 16 observed frequencies (O) and 16 expected frequencies (E). As in the case of the goodness of fit, calculate this χ^2 value. In our case, the computed χ^2 =131.76 as shown below: Each cell in the table below shows (O-E) ²/ (E)
Brand 1 Brand 2 Brand 3 Brand 4
Lower 8.20 21.49 14.08 23.36
Middle 0.27 2.86 4.00 0.33
Upper Middle 2.27 3.43 3.69 3.71
Upper 3.24 17.30 12.28 11.25
And there are 16 such cells. Adding all these 16 values, we get χ^2 =131.76
The critical value of χ^2depends on the degrees of freedom. The degrees of freedom = (the number of rows-1) multiplied by (the number of colums-1) in any contingency table. In our case, there are 4 rows and 4 columns. So the degrees of freedom =(4-1). (4-1) =9. At 5% level of significance, critical χ^2 for 9 d.f = 16.92. Therefore reject the null hypothesis and accept the alternative hypothesis.
The inference is that brand preference is highly associated with income level. Thus, the choice of the brand depends on the income strata. Consumers in different income strata prefer different brands. Specifically, consumers in upper middle and upper income group prefer premium brands while consumers in lower income and middle-income category prefer economy brands. The company should develop suitable strategies to position its detergent products. In the marketplace, it should position economy brands to lower and middle-income category and premium brands to upper middle and upper income category.
As a test for goodness of fit
The Chi-square test used with one sample is described as a "goodness of fit" test. It can help you decide whether a distribution of frequencies for a variable in a sample is representative of, or "fits", a specified population distribution. A number of marketing problems involve decision situations in which it is important for a marketing manager to know whether the pattern of frequencies that are observed fit well with the expected ones. The appropriate test is the χ^2 test of goodness of fit. The illustration given below will clarify the role of χ^2 in which only one categorical variable is involved.
Example: In consumer marketing, a common problem that any marketing manager faces is the selection of appropriate colors for package design. Assume that a marketing manager wishes to compare five different colors of package design. He is interested in knowing which of the five the most preferred one is so that it can be introduced in the market. A random sample of 400 consumers reveals the following:
Package color Preference by customers
Do the consumer preferences for package colors show any significant difference?
Solution: If you look at the data, you may be tempted to infer that Blue is the most preferred color. Statistically, you have to find out whether this preference could have arisen due to chance. The appropriate test statistic is the χ^2 test of goodness of fit.
Null Hypothesis: All colors are equally preferred.
Alternative Hypothesis: They are not equally preferred
Package color Observed Frequencies (O) Expected Frequencies (E) (〖O-E)〗^2 x^2=Σ(〖(O-E)〗^2/E
Red 70 80 100 1.250
Blue 106 80 676 8.450
Green 80 80 0 0.000
Pink 70 80 100 1.250
Orange 74 80 36 0.450
Total 400 400 11.400
Please note that under the null hypothesis of equal preference for all colors being true, the expected frequencies for all the colors will be equal to 80. Applying the formula
We get the computed value of chi-square χ^2 = 11.400
The critical value of χ^2 at 5% level of significance for 4 degrees of freedom is 9.488. So, the null hypothesis is rejected. The inference is that all colors are not equally preferred by the consumers. In particular, Blue is the most preferred one. The marketing manager can introduce blue color package in the market.
3) What is sampling and what are its uses.
Sampling is the process of selecting units (e.g., people, organizations) from a population of interest so that by studying the sample we may fairly generalize our results back to the population from which they were chosen
Probability sampling is based on random selection of units from a
population. In other words, the sampling process is not based on the
discretion of the researcher but is carried out in such a way that the
probability of every unit in the population of being included is the same.
For example, in the case of lottery, every individual has equal chance of
being selected. Some of the characteristics of a probability sample are :
i) each unit in the sample has some probability of entering the sample,
ii) weights appropriate to the probabilities are used in the analysis 01
the sample and
iii) the process of sampling is automatic in one or more steps of the
selection of units in the sample.
Probability sampling can be done tlwough different methods, each method
,having its own strengths and limitations. A brief account of these is give11
Non-probability sampling is based on the judgement of the researcher. The
guiding factors in non-probability sampling include the availability of the
units, the personal experience of the researcher and hidher convenience in
carrying out a survey. Since these samples are not prepared through random
sampling techniques, they are known as non-probability samples.
Depending on the technique used, non-probability samples are classified
into purposive, incidental and quota samples. A brief description of these
samples is given below.
Purposive sample .
A purposive sample is also known as a judgement sample. This type of.
sample is chosen because there are good reasons to believe that it is a
representative of the total population. This also reflects certain controls
identified as representative areas like a city, state or district; representative
characteristics of individuals like age, sex, marital status, etc; or types of
groups like school administrators, elementary school teachers, secondary.
school teachers, college teachers, etc.
A purposive sample differs from stratified random sample in that the actual
selection of the units to be included in the sample in each group is done
purposively rather than by random methods. For example, let us consider
the achievement level of housewives opting for distance education courses.
This approach comes in handy where it is necessary to include a very small
number of units in the sample. For example, for study of 'gifted' children,
the researcher, on the basis of hislher past experience, selects certain
individuals giving extra ordinary performance in school while excluding all
others from the sample.
The term incidental sample, also known as accidental sample, is applied to
samples that have been drawn because of the easy availability of units. An
investigator employed in the IGNOU may select learners enrolled for
Diploma in Distance Education while conducting a study on higher
education, as these learners are readily available and fulfil the conditions of
the study. But, neither of the two reasons may be of the investigator's
choice. Therefore, such casual groups rarely constitute random samples of
any definable population.
The merits of this procedure are mainly the convenience of obtaining units;
the ease of testing and completeness of the data collected. However, it is the
limitations that have defined population and no randomization has actually
been done. Therefore any attempt to arrive at generalised conclusion in
such cases will be erroneous and misleading.
Quota sanlple is another type of non-probability sample. It involves the
selection of sample units within each stratum or quota on the basis of the
judgement of the researcher rather than on calculable chance of the
individual units being included in the sample. Suppose a national survey has
to be done on the basis of quota sampling. The first step in quota sampling
would be to stratify the population region wise like rurallurban,
administrative districts etc. and then fix a quota of the sample to be selected.
In the initial state quota sampling is similar to stratified sampling. -
However, it may not necessarily employ random selection procedurd in the
initial stage in exactly the sanle way as probability sampling. The essential
difference between probability sampling and quota sampling lies in the I
selection of the final sampling units. The quota is usually determined by the
proportion of the groups. Suppose a researcher wants to study the attitude of
university teachers towards distance education. First of all, helshe may
stratify the university teachers in the category of sex and then as professors,
readers and lectures. Later, helshe may fix quotas for all these categories.
In this way, the quota sample would involve the use of strata but selection
within the strata is not done on a random basis.
The advantages of quota sampling are, its being less expensive,
conveaient, and more suitable in the case of missing or incomplete sampling
The non-probability samples are generally considered to be convenient
when the sample to be selected is small and the researcher wants to get some
idea of the population characteristics within a short time. In such cases, the .
primary objective of the researcher is to gain insight into the problem by
selecting only those persons who can provide miximum insight into the
However, the following are some inherent limitations of non-probability
i) No statistical theory has been devised to measure the reliability of
. results derived through purposive or other non-random samples.
Hence, no confidence can be placed in the data obtained from such
samples and results cannot be generalized for the entire population.
ii) The selective sampling based on convenience affects the variance
within the group as well as between the groups. Further, there is no
statistical method to determine the margin of sampling errors.
iii) Sometimes such samples are based on an obsolete frame which does
not adequately cover the population.
4) Is there any alternative formula to find the value of Chi-square?
Step 1 of 2
The Formula for Chi-Square Statistic
The chi-square statistic measures the difference between actual and expected counts in a statistical experiment. These experiments can vary from two way tables to multinomial experiments. The actual counts are from observations, the expected counts are typically determined from probabilistic or other mathematical models.
In the above formula we are looking at n pairs of expected and observed counts. The symbol ek denotes the expected counts, and fk denotes the observed counts. To calculate the statistic, we do the following steps:
Calculate the difference between corresponding actual and expected counts.
Square the differences from the previous step, similar to the formula for standard deviation.
Divide every one of the squared difference by the corresponding expected value.
Add all of these numbers together to give us our chi-square statistic.
The result is a nonnegative number that tells us how much different the actual and expected counts are. If we compute that χ2 = 0, then this indicates that there are no differences between any of our observed and expected counts.
An alternate form of the equation for the chi-square statistic uses summation notation in order to write the equation more compactly. This is seen in the second line of the above equation.
To see how to compute a chi-square statistic using the formula, suppose that we have the following data from an experiment:
Expected: 25 Observed: 23
Expected: 15 Observed: 20
Expected: 4 Observed: 3
Expected: 24 Observed: 24
Expected: 13 Observed: 10
Next, compute the differences for each of these. Because we will end up squaring these numbers, you may subtract them in any order. Staying consistent with our formula, we will subtract the observed counts from the expected ones:
25 – 23 = 2
15 – 20 =-5
4 – 3 = 1
24 – 24 = 0
13 – 10 = 3
Now square all of these differences: and divide by the corresponding expected value:
22/25 = 0 .16
(-5)2/15 = 1.6667
12/4 = 0.25
02/24 = 0
32 /13 = 0.5625
Finish by adding the above numbers together: 0.16 + 1.6667 + 0.25 + 0 + 0.5625 = 2.693
Further work involving hypothesis testing would need to be done to determine what significa