Explain the sample distribution of mean with reference to some numerical example. Illustrate the practical implications of the Central Limit Theorem (CLT) in inferential statistics.

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Suresh Chand · Accepted Answer

A sampling distribution of a statistic is a type of probability distribution created by drawing many random samples of a given size from the same population. These distributions help you understand how a sample statistic varies from sample to sample. Let us consider the sample distribution of sample mean x̄. Let us suppose that a random sample size n is taken from a normal population with mean x̄ and variance σ2. Let x1, x2, x3 . . . . xn is a random sample (without replacement) of size n from a finite population of size N, then E(x̄) = μ mean value of the sample distribution of sample mean is equal to population mean and its varience is given by (V(bar{X}) = frac{σ^2 (N-n)}{n (N-1)}) which measures the variability of sample mean. In the same way its sandard error will be S.E. (x̄) = (frac{σ}{sqrt{n}} sqrt{frac{(N-n)}{N - 1}}) Numerical Example: A sample of size 36 is drawn from a population consisting of 196 units. If the population standard deviation is 7, find the standard error of the sample mean. Solution: Population size(N) = 196 Sample size(n) = 36 Population SD(σ) = 7 Now, Sandard Error is given by S.E.(x̄) = (frac{σ}{sqrt{n}} sqrt{frac{(N-n)}{N - 1}}) = (frac{7}{sqrt{36}} times sqrt{frac{196 - 36}{196 - 1}}) = 1.0568 Central Limit Theorem: The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30. Practical Implication of CLT: One of the most common applications of central limit theorem is in election polls. To calculate the percentage of persons supporting a candidate which are seen on news as confidence intervals. Economists often use the central limit theorem when using sample data to draw conclusions about a population. For example, an economist may collect a simple random sample of 50 individuals in a town and use the average annual income of the individuals in the sample to estimate the average annual income of individuals in the entire town. Biologists use the central limit theorem whenever they use data from a sample of organisms to draw conclusions about the overall population of organisms. For example, a biologist may measure the height of 30 randomly selected plants and then use the sample mean height to estimate the population mean height. Manufacturing plants often use the central limit theorem to estimate how many products produced by the plant are defective. For example, the manager of the plant may randomly select 60 products produced by the plant in a given day and count how many of the products are defective. He can use the proportion of defective products in the sample to estimate the proportion of all products that are defective that are produced by the entire plant.

Explain the sample distribution of mean with reference to some numerical example. Illustrate the practical implications of the Central Limit Theorem (CLT) in inferential statistics.

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Explain the sample distribution of mean with reference to some numerical example. Illustrate the practical implications of the Central Limit Theorem (CLT) in inferential statistics.

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