Question

A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. He wants to construct the 95% confidence interval with a maximum error of 0.09 reproductions per hour. Assuming that the mean is 6.6 reproductions and the variance is known to be 4.84, what is the minimum sample size required for the estimate? Round your answer up to the next integer.

Answer 1

The minimum sample size is
`n = 2295`

Explanation:

From the question we are told that

The margin of error is
`E = 0.09`

The sample mean is
`\= x = 6.6`

The variance is
`\sigma^2 = 4.84`

Generally the standard deviation is mathematically represented as

`\sigma = √(\sigma^2)`

=>
`\sigma = √(4.84 )`

=>
`\sigma = 2.2`

From the question we are told the confidence level is 95% , hence the level of significance is

`\alpha = (100 - 95 ) \%`

=>
`\alpha = 0.05`

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

`Z_{(\alpha )/(2) } =  1.96`

Generally the sample size is mathematically represented as

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

=>
`n = [(1.96  *  2.2 )/(0.09) ] ^2`

=>
`n = 2295`