C3 Coursework - different methods of solving equations.
C3 Coursework - Numerical Solutions Decimal Search There a numerous ways to solve a problem and in finding the unknown. Some methods give you the exact and precise answer but usually are harder and more complex. The Decimal search method enables you to get a very close approximate to the real solution but more easily. The way this method works is by looking between two numerical values (for example 1 and 2) and then As a demonstration in applying this method, I will be attempting to solve this equation using the Decimal Search method and going through the method step by step: Below is what this function looks when plotted on a graph: We know that the solution for F(x) = 0 is the point on the X axis where the sign changes from a positive to a negative. So if we zoom in a little bit further, from this graph we can tell where the solution lies, somewhere between 0 and 10 Now that we know the solution is roughly between these two values, I will use excel to solve the problem with firstly taking increments in x, the size of 1. So when I substitute the incremented values of x between -10 and 0 into the equation, I get the following results: x F(x) -1 20 -2 9 -3 0 -4 -13 -5 -56 -6 -125 -7 -226 -8 -365 -9 -548 You can tell that the sign changed between -3 and -4. So I set these as my initial values. The fact that the solution lies between -3 and -4 can
The heights of 16-18 year old young adults varies between males and females. My prediction is that the majority of males are taller than females.
Statistics coursework S1 Task A: Measurements The heights of 16-18 young adults varies between males and females. My prediction is that the majority of males are taller than females. I have decided to take a sample of 100 16-18 young adults from Havering Sixth Form College and measure their heights with a tape measure. My two populations will be male and females and I will obtain 50 male heights and 50 female heights. I have chosen to take a sample of 100 because small sample sizes may give inaccurate results whereas a larger sample size may be impractical and time consuming. Therefore I feel a sample size of 100 will give a efficient set of results and a meaningful conclusion could be deduced. To insure that the data I collected wasn't biased I didn't look at each persons height to ensure that the data I collected was even for both populations and within both populations. Also because height isn't affected by whether you are athletic or not or whether you eat lots or who you are friends with, the data I collected cant be bias for that aspect of my data collection and also I didn't collect my data at a modeling studio, of which nearly everyone there would be over 6ft and would make my data collection bias and unfair. So overall the data I collected wasn't bias and was accurate to use for my investigation. I decided to investigate the difference in heights between
Estimate a consumption function for the UK economy explaining the economic theory and statistical techniques you have used.
Estimate a consumption function for the UK economy explaining the economic theory and statistical techniques you have used. Consumption has been considered as the most important single element in aggregate demand, accounting for almost 66% of GDP in 1989. Therefore, it is essential that the level of consumption be predicted accurately, for even a small percentage error may lead to a large absolute error. Another reason why consumption such important is that the marginal propensity of consume is one of the factor which is used to determine the size of the multiplier. This will influence the changes of investment and government spending. Moreover, as saving ratio, another factor that used to predict the behavior of consumers, has fluctuated rapidly during these years, so the consumption become more important. Thus, many economists attempt to develop many theories and equations to predict consumers' expenditure. Therefore, in this project, there are two main theories and their equations will be given. Some data from Economic Trends Annual Supplement and graphs will be used to estimate the performance of the model on the basis of some econometric tests. Keynes, John Maynard introduced consumption function in 1936. In The General Theory of Employment Interest and Money, Keynes pointed out that " We should therefore define what shall call the propensity to consume as the
Numerical solutions of equations
A2 Mathematics Coursework C3 Year 12 Numerical solutions of equations Solving 0 = x5+x-5 using the "Change Of Sign" Method The method I will use to solve 0 = x5+x-5 is the Change of Sign Method involving the Decimal Search method. I have drawn this graph using the Autograph Software, and the print screen of this is below: From my graph above, I can see that the root of this equation is between x =1 and x = 1.5. The table of x values and f(x) values is shown below. I can work out the f(x) values by substituting the x-values into the equation. x .1 .2 .3 .4 .5 f(x) -3 -2.28949 -1.31168 0.01293 .77824 4.09375 From my table of values above, it is clear that the change of sign from negative to positive occurs between x = 1.2 and x = 1.3. So, I can narrow these values down further to find another change of sign. x f(x) .21 -1.19626 .22 -1.07729 .23 -0.95469 .24 -0.82837 .25 -0.69824 .26 -0.56420 .27 -0.42616 .28 -0.28403 .29 -0.13769 .30 0.01293 I can see that the change of sign is between x = 1.29 and x = 1.30. x f(x) .291 -0.12283 .292 -0.10792 .293 -0.09296 .294 -0.07797 .295 -0.06293 .296 -0.04784 .297 -0.03271 .298 -0.01754 .299 -0.00233 .300 0.01293 The change of sign is in the interval [1.299, 1.300] x f(x) .2991 -0.000805 .2992 0.000720 .2993 0.002244 The root of this equation lies in the interval
data handling
Introduction I have chosen three categories, which are the height, hand span and the shoe size from the data. My first hypothesis is that taller people will have bigger feet. My second hypothesis is that taller people will have bigger hand spans. I think that there will be a positive correlation between the height with shoe and hand size because it's common sense. Most people I see on the streets have big hands and feet when they are tall. I think that both correlations will be very high because of this. I picked a sample of 50 people because 50 out of about 261 pupils will be just about 20% of the people. It uses some of the people and 50 is a nice number to work with. I picked the sample of 50 by using stratified sampling since a random choice is not representative. The sample might contain all 50 boys or all 50 girls might it is random. Also, a stratified sample is more representative because it would be accurate by being based on the information, but it is not perfect. The data that I needed to stratify are the date of births and the gender of the pupils because older pupils are normally taller. Months Boys Girls Total Amount September 1 9 20 October 4 6 20 November 7 3 20 December 9 7 26 January 3 6 9 February 7 0 27 March 5 7 32 April 9 8 7 May 9 1 20 June 2 3 25 July 9 6 25 August 4 6 0 29 32 261 I counted
The Gradient Function
The Gradient Function Aim: To find the gradient function of curves of the form y=axn. To begin with, I should investigate how the gradient changes, in relation to the value of x. Following this, I plan to expand my investigation to see how the gradient changes, and as a result how a changes in relation to this. Method: At the very start of the investigation, I shall investigate the gradient at the values of y=xn. To start with, I shall put the results in a table, but later on, as I attempt to find the gradient through advanced methods, a table may be unnecessary. As I plot the values of y=x2, this should allow me to plot a line of best fit and analyze, and otherwise evaluate, the relationship between the gradient and x in this equation. I have begun with n=2. After analyzing this, I shall carry on using a constant value of "a" until further on in the investigation, and keep on increasing n by 1 each time. I shall plot on the graphs the relative x values and determine a gradient between n and the gradients. Perhaps further on in the investigation, I shall modify the value of a, and perhaps make n a fractional or negative power. Method to find the gradient: These methods would perhaps be better if I demonstrated them using an example, so I will illustrate this using y=x2. This is the graph of y=x2. I will find out the gradient of this curve, by using the three methods -