Question

We wish to estimate the population mean of a variable that has standard deviation 70.5. We want to estimate it with an error no greater than 5 units with probability 0.99. How big a sample should we take from the population? What happens if the standard deviation and the margin of error are both doubled?

Answer 1

a) The large sample size 'n' = 1320.59

b) If the standard deviation and the margin of error are both doubled also the sample size is not changed.

Explanation:

Explanation:-

a)

Given data the standard deviation of the population

σ = 70.5

Given the margin error = 5 units

We know that the estimate of the population mean is defined by

that is margin error =
`(z_(\alpha ) S.D )/(√(n) )`

`M.E = (z_(\alpha ) S.D )/(√(n) )`

cross multiplication , we get

`M.E (√(n) ) = z_(\alpha ) S.D`

`√(n) = (z_(\alpha ) S.D )/(M.E )`

`√(n) = (2.578 X 70.5)/(5) }`

√n = 36.34

squaring on both sides , we get

n = 1320.59

b) The margin error of the mean

`√(n) = (z_(\alpha ) S.D )/(M.E )`

the standard deviation and the margin of error are both doubled

√n = zₓ2σ/2M.E

√n = 36.34

squaring on both sides , we get

n = 1320.59

If the standard deviation and the margin of error are both doubled also the sample size is not changed.