- The distribution of the sample means $x$ over $\bar{x}$ will, as the sample size increases, approach a normal distribution.
- The distribution of the sample data will approach a normal distribution as the sample size increases.
- The standard deviation of all sample means is the population standard deviation divided by the square root of the sample size.
- The mean of all sample means is the population mean $\mu$.
This question aims to choose the correct statement out of the given four statements regarding the conclusion of the Central Limit Theorem.
The Central Limit Theorem is a statistical concept stating that there will be normally distributed samples with a sample mean approximately equal to the population mean if a large sample size has a finite variance. To put it another way, add up the means from all of the samples, and find the average which will be equal to the population mean. Likewise, if all the standard deviations in the sample are averages, the population standard deviation will be obtained.
This is true yet if the population taken is skewed or normal, as long as the sample size is large enough (generally $n \geq 30$). The theorem remains true also for samples less than $30$ if the population is normal. This is also true even though the population is binomial, as long as $min(np, n(1-p))\geq 5$, where $n$ is the sample size and $p$ is the population’s success probability. This implies that one can use the normal probability model to measure unpredictability when inferring population means from sample means. The Central Limit Theorem applies to almost all probability distributions. However, there are some exclusions. For instance, suppose that the variance of the population is finite. This theorem is also applicable to variables that are independent and identically distributed. It can also be used to determine how large a sample is required.
Expert Answer
The statement, “The distribution of the sample data will approach a normal distribution as the sample size increases,” is not the conclusion for Central Limit Theorem.
The reasons for the other given statements to be correct are:
As the sample size increases, the distribution of the sample mean approaches normality. The expected value of all sample means is equal to the population mean, and the standard deviation of all sample means is the ratio of the population standard deviation to the square root of sample size.
The sample mean distribution tends to normal distribution with the increase in sample size.
The population standard deviation divided by the square root of the sample size equals the standard error of all the sample means.
Also, the population mean equals the expected value of all the sample means.
And the reason for the given incorrect statement is:
Hence, by the Central Limit Theorem, the sample data distribution will not tend to a normal distribution with the increase or decrease in the sample size. But on the other hand, the sample mean average will.
Example
Find the sample mean and standard deviation if the ages of the female population are normally distributed with a mean of $60$ and a standard error of $20$ when the sample of $40$ females is taken.
Solution
Given:
$\mu=60$, $\sigma=20$ and $n=40$
So that:
$\mu_{\bar{x}}=\mu=60$
$\sigma_{\bar{x}}=\dfrac{\sigma}{\sqrt{n}}$
$=\dfrac{20}{\sqrt{40}}$
$\sigma_{\bar{x}}=3.162$