Let X ∼ N(µ, σ) and X¯ be samp

Let X ∼ N(µ, σ) and X¯ be sample mean from a random sample of9.

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

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

Answer:

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)

Confidenceintervals

The condition ofconfidence 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 thePopulation 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 confidenceintervals 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 thenthe standard error of mean will be multiplied by

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

  

     


 
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