Solved Examples on Central Limit Theorem
Example 1. The male population’s weight data follows a normal distribution. It has a mean of 70 kg and a standard deviation of 15 kg. What would the mean and standard deviation of a sample of 50 guys be if a researcher looked at their records?
Solution:
Given: μ = 70 kg, σ = 15 kg, n = 50
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 70 kg
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex] = 15/√50
⇒ [Tex]\sigma _{\overline{x}} [/Tex] ≈ 2.1 kg
Example 2. A distribution has a mean of 69 and a standard deviation of 420. Find the mean and standard deviation if a sample of 80 is drawn from the distribution.
Solution:
Given: μ = 69, σ = 420, n = 80
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 69
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex]
⇒ [Tex]\sigma _{\overline{x}} [/Tex]= 420/√80
⇒ [Tex]\sigma _{\overline{x}} [/Tex] = 46.95
Example 3. The mean age of people in a colony is 34 years. Suppose the standard deviation is 15 years. The sample of size is 50. Find the mean and standard deviation of the sample.
Solution:
Given: μ = 34, σ = 15, n = 50
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 34 years
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex]
⇒ [Tex]\sigma _{\overline{x}} [/Tex]= 15/√50
⇒ [Tex]\sigma _{\overline{x}} [/Tex] = 2.12 years
Example 4. The mean age of cigarette smokers is 35 years. Suppose the standard deviation is 10 years. The sample size is 39. Find the mean and standard deviation of the sample.
Solution:
Given: μ = 35, σ = 10, n = 39
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 35 years
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex] = 10/√39
⇒ [Tex]\sigma _{\overline{x}} [/Tex] = 1.601 years
Example 5. The mean time taken to read a newspaper is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of size 70. Find its mean and standard deviation.
Solution:
Given: μ = 8.2, σ = 1, n = 70
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 8.2 minutes
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex] = 1/√70
⇒ [Tex]\sigma _{\overline{x}} [/Tex] = 0.11 minutes
Example 6. A distribution has a mean of 12 and a standard deviation of 3. Find the mean and standard deviation if a sample of 36 is drawn from the distribution.
Solution:
Given: μ = 12, σ = 3, n = 36
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 12
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex] = 3/√36
⇒ [Tex]\sigma _{\overline{x}} [/Tex] = 0.5
Example 7. A distribution has a mean of 4 and a standard deviation of 5. Find the mean and standard deviation if a sample of 25 is drawn from the distribution.
Solution:
Given: μ = 4, σ = 5, n = 25
As per the Central Limit Theorem, the sample mean is equal to the population mean.
Hence, [Tex]\mu _{\overline{x}} [/Tex] = μ = 4
Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex]
⇒ [Tex]\sigma _{\overline{x}} [/Tex]= 5/√25
⇒ [Tex]\sigma _{\overline{x}} [/Tex] = 1
Articles related to Central Limit Theorem:
Central Limit Theorem
The Central Limit Theorem in Statistics states that as the sample size increases and its variance is finite, then the distribution of the sample mean approaches normal distribution irrespective of the shape of the population distribution. The query that how much the sample size should increase can be answered that if the sample size is greater than 30 then the statement of the Central Limit Theorem holds.
The central limit theorem posits that the distribution of sample means will invariably conform to a normal distribution provided the sample size is sufficiently large. This holds regardless of the underlying distribution of the population, be it normal, Poisson, binomial, or any alternative distribution.
In this article on the Central Limit Theorem, we will learn about the Central Limit Theorem definition, Central Limit Theorem examples, Central Limit Theorem Formulas, proof of the Central Limit Theorem, and Central Limit Theorem applications.
Table of Content
- Central Limit Theorem in Statistics
- Central Limit Theorem Definition
- Central Limit Theorem Formula
- Central Limit Theorem Proof
- Central Limit Theorem Examples
- Assumptions of the Central Limit Theorem
- Steps to Solve Problems on Central Limit Theorem
- Central Limit Theorem Applications
- Solved Examples on Central Limit Theorem
- Summary – Central Limit Theorem in Statistics