Question

Let X ∼ N(µ, σ) and X¯ be sample mean from a random sample of 9.

Suppose you draw a random sample of 9, calculate an interval ¯x ± 0.5σ where σ is the population standard deviation of X, and then check whether µ, the population mean, is contained in the interval or not.

If you repeat this process 100 times, about how many time do you think µ is contained in X¯ ± 0.5σ. Explain why. (Hint: What is the value of z-multiplier of x¯ ± 0.5σ?)

Answer 1

Explanation:-

a)

Step(i):-

Given  X¯ be sample mean from a random sample

Given random sample size 'n' =9

Given data the confidence intervals are    ...(a)

Confidence intervals

The condition of confidence intervals for 'µ' is given by

..(b)

Comparing (a) and (b) equations

Given sample size 'n' =9

The value of z- multiplier = 1.5

The confidence intervals for 'µ'

Step(ii) :-

b) Given the sample size is n=100

Given data the confidence intervals are      ...(a)

The condition of confidence intervals for the Population mean 'µ' is given by

..(b)

Comparing (a) and (b) equations

cancellation'  ' on both sides , we get

Given sample size 'n' =100

The value of z- multiplier is 5 of sample size n=100

The confidence intervals are  

Step(iii):-

The standard error of the mean

Given sample size 'n' = 9

    

Given the repeated this process is 100 times so

If n =9 is incresed to

Again the standard error of the mean

  

Now From (ii)

Here multiply Z  multipliers of

  

  

Conclusion:-

The standard error of the mean

Therfore the sample size is increased from 9 to 100 times then the standard error of mean will be multiplied by

Hence 3.33 times of Population mean is contained in X¯ ± 0.5σ.