What is the difference between confidence interval and the width of confidence interval? Let X_1, ..., X_n be n random samples of normal distribution X ~ N(mu, 4) where mu is unknown.

Kelton Bailey

Kelton Bailey

Answered question

2022-09-29

What is the difference between confidence interval and the width of confidence interval?
Let X 1 , . . . , X n be n random samples of a normal distribution X N ( μ , 4 ) where μ is unknown. Write down the 95% confidence interval of μ in terms of n. What sample size n is required such that the width of this confidence interval is barely larger that 0.5?

Answer & Explanation

Roger Clements

Roger Clements

Beginner2022-09-30Added 4 answers

Step 1
The required confidence interval of μ is:
( x ¯ z α / 2 σ n , x ¯ + z α / 2 σ n ) = ( x ¯ 1.96 2 n , x ¯ + 1.96 2 n )
Step 2
The width of the confidence interval is:
( x ¯ + 1.96 2 n ) ( x ¯ 1.96 2 n ) = 7.84 n ,
which must be slightly larger than 5:
7.84 n > 5 n < 1.568 n < 2.5 n = 2
Note that σ n is called Standard Error, while z α / 2 σ n is called Margin of Error (or Sampling Error). Indeed, the ME is half of the width of confidence interval.

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