beljuA

2021-10-20

The population mean may be reliably estimated using a statistic calculated from data recorded from randomly selected individuals.’ Explain this statement with reference to sampling distributions and the central limit theorem.

Alara Mccarthy

Step 1
Given information:
The population mean may be reliably estimated using a statistic calculated from data recorded from selected individuals.
Step 2
Sampling distribution of sample mean:
Let a particular characteristic of a population is of interest in a study. Denote µ as the population mean of that characteristic and σ as the population standard deviation of that characteristic.
Now, it is not always possible to study every population unit. So, a sample of size n is taken from the population.
Let X denotes the random variable that measures the particular characteristic of interest. Let, ${X}_{1},{X}_{2},\dots ,{X}_{n}$ be the values of the random variable for the n units of the sample from a normal distribution.
Then, the sample mean has a sampling distribution that has population mean or expected value same as that of X and population standard deviation, called the standard error, which is the population standard deviation of X divided by the square root of the sample size n. The parameters are as follows:
$E\left(\stackrel{―}{X}\right)=E\left(\frac{1}{n}\sum {X}_{i}\right)$
$=\mu$
$SD\left(\stackrel{―}{X}\right)=\frac{\sigma }{\sqrt{n}}$.
Step 3
Central limit theorem:
Central limit theorem states that, the shape of the sampling distribution of sample mean will be approximately normal, if the sample size is large and the standard deviation of the population is finite.
If ${X}_{1},{X}_{2},\dots ,{X}_{n}$ are independent and identically distributed (iid) random variables with finite mean, $E\left({X}_{i}\right)=\mu$ and finite variance $V\left({X}_{i}\right)={\sigma }^{2}$, then Z will converge to the standard normal distribution when the sample size, n tends to infinity, where:
$Z=\frac{\stackrel{―}{X}-\mu }{\frac{\sigma }{\sqrt{n}}}\sim N\left(0,1\right)$
As a rule of thumb if the selected sample size is greater than 30, then the sample mean will be approximately normally distributed.
Thus, the sample mean will be unbiased estimator of the population mean.

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