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’
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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
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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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