Question

. For n = 2 look at the smallest and largest sample means. How close is each of these sample means to the population mean? Use the formula x − . This difference is called the sampling error.

Do the same for n = 3 and n = 4. What do you notice as n gets larger?

Answer 1

Suppose the population consist of elements { 4,5,6,7}

Considering n=2, we take sample 1 :  (5,4) have sample mean =4.5

Sample 2 : (6,4) have sample mean = 5

Sample 3 : ( 6,5) have sample mean= 5.5

sample 4: (6,7) have sample mean = 6.5

Sample 5: (5,7) have sample mean = 6

Sample 6: (4,7) have sample mean= 5.5

Largest sample mean = 6.5

Smalles Sample Mean = 4.5 , given the set of observation

We know that, Mean of Sample Mean = Population mean according to Central Limit Theorem

Hence Population mean = (4.5 + 5+ 5.5 + 6.5+ 6 + 5.5 ) /6  

= 33 / 6

= 5.5

Therefore, Sampling error = 1 unit

now for n = 3, we take sample 1 : (4,5,6) have sample mean= 5

sample2 : (4,5,7) have sample mean= 5.33

Sample 3 : ( 5,6,7) have sample mean = 6

Sample 4  : (4,6,7) have sample mean = 5

Here largest sample mean = 6

Smallest sample Mean = 5

Population mean = (5 + 5.33 + 6 + 5 ) /4

= 5.33

Largest Sampling error = (6 - 5.33 )

= 0.67

when n =4 , sample 1: (4,5,6,7) have sample mean = 5.5

Population mean = (4+4+6+7) / 4 = 22/4 = 5.5

Here sampling error = 0

Hence from the above it can observed that , as Sample size increases, Sampling distribution of mean tends to normality or closer to normal distribution and it justifies the Central Limit Theorem .