Question

[9] X is a Gaussian random variable with variance 0.25. The mean of X is estimated by taking the sample mean of independent samples of X. If the mean needs to be estimated within 0.01 from the actual mean with a confidence coefficient of 0.99, find the minimum number of samples needed in the estimation.

Answer 1

The minimum number of samples required is
`n = 16641`

Explanation:

From the question we are told that

The variance is
`\sigma^2 = 0.25`

The margin of error is
`E = 0.01`

From the question we are told the confidence coefficient is 0.99 , hence the level of significance is

`\alpha = (1 - 0.99 ) \%`

=>
`\alpha = 0.01`

Generally from the normal distribution table the critical value of
`(\alpha )/(2)` is

`Z_{(\alpha )/(2) } =  2.58`

Generally the standard deviation is

`\sigma =√(\sigma^2)`

=>
`\sigma =√(0.25)`

=>
`\sigma =0.5`

Generally the sample size is mathematically represented as

`n = [\frac{Z_{(\alpha )/(2) } *  \sigma }{E} ] ^2`

=>
`n = [\frac{2.58 } *  0.5 }{0.01 } ] ^2`

=>
`n = 16641`