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Introduction to Central Limit Theorem and Steps to Check Whether an Equation Is Linear

Theorem

Central Limit Theorem

Statement of Central Limit Theorem: Suppose large random samples are considered from the population with replacement consisting of mean μ and standard deviation σ, then the sample means’ distribution is normally distributed. The theorem is valid irrespective of whether the population is skewed or normal. The only criteria is that the sample should be ≥ 30. Central Limit Theorem is accurate provided that the population approaches a binomial distribution if the minimum [f (np, nq) > 5], where, n denotes the sample size, p is the probability of success of the population and 1 – p = q since p + q = 1.

The computation of sample means and sample standard deviation that are obtained from the population are as follows:

sample mean and standard deviation formula

Few illustrations of Central Limit Theorem are discussed below

1] Suppose a coin is flipped many times, the probability of getting a sure number of heads will approach a normal distribution, provided the mean = half of the total number of flips of the coin. It will equate to a normal distribution at the limit of an infinite number of flips.

2] The total distance covered in a morning or an evening walk will approach towards a normal distribution.

3] A single quantity can be weighted as the average of several small effects in the case of electronic noise or exam grades.

The process involved in obtaining the solution of the problem based on the Central Limit Theorem that consists of “>” “<” or “=” are as follows.

1] From the given problem, the known quantities such as mean, population and sample size, standard deviation and the values associated with the respective sign is to be identified.

2] A graph with a mean as its centre is to be drawn.

3] Calculate the z score using the formula:

z-score formula

4] Obtain the z-value from the z-table.

5] The various cases associated with different signs are

a] Central Limit Theorem with “>”: 0.5 – [z-score]

b] Central Limit Theorem with “<”: 0.5 + [z-score]

c] Central limit theorem that involves “between”.

Step 3 to be repeated.

6] The values of x{bar} and z-value has to be found and is expressed in percentage by converting it from decimal to percentage in the end.

Uses of Central Limit Theorem

1] To find a range of values that includes the population mean, sample mean is used.

2] Since there is an increase in the samples that are taken from the population, it leads to a decrease in the sample mean deviation which in turn helps in calculating the population mean more accurately.

3] Central Limit Theorem is implemented in political election polls to find the number of voters in percentage who are in favour of a particular candidate as confidence intervals which are shown by the news channels.

Check more JEE questions on topics like sets, relations and functions, permutation and combination, mathematical equations and inequalities, three dimensional geometry and many more at BYJU’S. This page answers the question – is an equation linear. For more information on linear equations, equations of a line, different forms of linear equations, solutions to linear equations, please refer to the above page.