Chebyshev’s Theorem and the Empirical Rule

A bell curve is perfectly symmetrical with respect to a vertical line through its peak and is sometimes called a Gauss curve or a normal curve. The second shape a scatter diagram may have is anything but a normal curve as in the next drawing: We can do a lot of good statistics with the normal curve, but virtually none with any other curve. Let us assume that we have recorded the 1000 ages and computed the mean and standard deviation of these ages. Assuming the mean age came out as 40 years and the standard deviation as 6 years we can do the following predictions. Chebyshev’s Theorem In the case of a scatter diagram that seems to be anything but a normal curve, all we can go by is Chebyshev’s theorem. This very important but rarely used theorem states that in those cases where we have a non-normal distribution, the following can be said abut the individual data, which in this case are the ages: †¢ At least 75% of all the ages will lie in the range of [pic]. In our case this means that at least 75% of the people will have an age in the range of [pic] years which simplifies to a range of 28 to 52 years. †¢ At least 88. 9% of all the ages will lie in the range of [pic]. In our case this means that at least 88. 9% of the people will have an age in the range of [pic] years which simplifies to a range of 22 to 58 years. †¢ At least 93. 75% of all the ages will lie in the range of [pic]. In our case this means that at least 93. 75% of the people will have an age in the range of [pic] years which simplifies to a range of 16 to 64 years. †¢ At least 96% of all the ages will lie in the range of [pic]. In our case this means that at least 96% of the people will have an age in the range of [pic] years which simplifies to a range of 10 to 70 years. At least 97. 2% of all the ages will lie in the range of [pic]. In our case this means that at least 97. 2% of the people will have an age in the range of [pic] years which simplifies to a range of 4 to 76 years. How can we calculate these percentages? To calculate the 75%, the 88. 9%, the 93. 75%, etc, we look at the number of standard deviations in the respective intervals. The 75% goes together with me an  ± 1 standard deviation, the 88. 9% with mean  ± 2 standard deviations, the 93. 75% with mean  ± 3 standard deviations, and the 96% with mean  ± 4 standard deviations. In general you can say that the percentage of people with an age in the range of mean  ± k standard deviations can be found by calculating the value of the quantity [pic] and then converting that into a percentage. Summarizing the above we get the following table: |Interval |k |[pic] |% | |[pic] |2 |[pic] |75 | |[pic] |3 |[pic] |88. 9 | |[pic] |4 |[pic] |93. 75 | |[pic] |5 |[pic] |96 | [pic] |6 |[pic] |97. 2 | Do we have to restrict ourselves to whole numbers as values for k? No, we may take any value for k as long as it larger than 1. For instance, for k = 2. 5 we get the result that [pic] in the interval [pic] years Example 1: Students Who Care is a student volunteer program in which college students donate work time in community centers for homeless people. Professor Gill is the faculty sponsor for this student volunteer program. For several years Dr. Gill has kept a record of the total number of work hours volunteered by s student in the program each semester. For students in the program, for each semester the mean number of hours was 29. 1 hours with a standard deviation of 1. 7 hours. Find an interval for the number of hours volunteered in which at least 88. 9% of the students in this program would fit. Solution: From the table above we see that a percentage of 88. 9 will coincide with an interval of [pic] hours. This can be rewritten as an interval from 24 to 34. 2 hours volunteered each semester. Example 2: The East Coast Independent News periodically runs ads in its own classified section offering a month’s free subscription to those who respond. This way management can get a sense about the number of subscribers who read the classified section each day. Careful records have been kept over a period of 2 years. The mean number of responses was 525 with a standard deviation of 30. What is the smallest percentage of responses in the interval between 375 and 675? Solution: The difference between the mean of 525 and the upper limit of this interval is 150. This is 5 standard deviations since[pic]. The same is true for the difference between the mean and the lower limit of this interval. According to the table above this coincides with 96%. The Empirical Rule When the data values seem to have a normal distribution, or approximately so, we can use a much easier theorem than Chebyshev’s. The empirical rule states that in cases where the distribution is normal, the following statements are true: †¢ Approximately 68% of the data values will fall within 1 standard deviation of the mean. †¢ Approximately 95% of the data values will fall within 2 standard deviations of the mean. Approximately 99. 7% of the data values will fall within 3 standard deviations of the mean. Example 3: The average salary for graduates entering the actuarial field is $60,000. If the salaries are normally distributed with a standard deviation of $5000, then what percentage of the graduates will have a salary between $50,000 and $70,000? Solution: Both $50,000 and $70,000 are $10,000 away from the mean of $60,000. This is two standard deviations away from the mean, so 95% of the graduates will have a salary in this interval. [pic] Age No of people [pic]

Highlight problems and issues trade union face in the UK and recommend Essay

Highlight problems and issues trade union face in the UK and recommend practical solutionsto solve them - Essay Example Failure to come on an agreement on the negotiations made, the trade unions are entitled to strike. Capitalism is a form of governance where, the land, labor and other factors of production are privately owned states Clarke (1978). Trade unions can be defined from this perspective as the representatives of the employees’ terms of employment to employers and government from a given policy which is based on the person who privately owns the union. The trade unions here are not government controlled but privately controlled. Trade union from a radical perspective can be defines as the representation of employees from the industrial and commercial sectors only. These are the unions that only focus on the two business sectors but they are not necessarily privately owned. Marxism emphasises on the workers’ rights to enjoy the fruits of their labor. It thus defines trade unions as organs which represent work towards ceasing employees’ harassment at work place and ensuring they enjoy the benefits and results that come from their work. Lionel (1968) states that in the UK, most unions belong to the Trades Union Congress, which represents many unions, and hence has more power over issues effecting workers nationally. Therefore trade unions act as a voice for the employees and often take on the role of challenging and updating contracts and conditions for members and also protect the members from harassment and employment related legal issues. Rosen (1969) states the main sources of power of the trade unions as the government, employers, and employees. These are the key sources of trade unions as they are the determinants of the representatives in the trade unions and the legal issues that it should abide to. Stewart (1986) argues that if a union does not have a good collective bargaining style, then it will lose its members who are the employees. On the other hand, the employers and the

Business Activities, Employment and Inflation Term Paper

Business Activities, Employment and Inflation - Term Paper Example If the injections of an economy (J) are not equal to the withdrawals (W), then there is a clear disequilibrium in the economy. The only factors that bring this equilibrium back into line is the change in the national income (GDP) and the levels of employments (Sloman and Sutcliffe). To understand the above figure better, lets consider that the economy is faced with a state of equilibrium, i.e. the levels of withdrawals and the level of injections is the same. If there is an increase in the injections, and the firms aim at investing more into the company, then the aggregate demand, i.e. Cd will also be higher. Hence to meet up with this demand, the firms will also need to increase the labor and other resources which would in turn lead to higher levels of incomes for the households (Y) (Sloman and Sutcliffe). With an increase in the income of the households, there will be an increase in the expenses as well, which in turn will lead the firms to also sell higher. Higher sales will mean the firms need to produce higher which again would mean more labor, and other resources (Mankiw). This is a multiplied affect that will continue to go on within the economy. This effect is referred to as, ‘Multiplier Effect’ and is defined as, ‘an initial increase in aggregate demand of $Xm leads to an eventual rise in national income that is greater than $Xm’ (Sloman and Sutcliffe). This follows the principle of ‘Cumulative Causation’, which can be defined as, ‘An initial event can cause an ultimate effect which is much larger’ (Sloman and Sutcliffe).