Question

A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. The mean is found to be 13.6 reproductions and the population standard deviation is known to be 1.9. If a sample of 189 was used for the study, construct the 85% confidence interval for the true mean number of reproductions per hour for the bacteria. Round your answers to one decimal place.

Answer 1

The confidence interval is
`13.4< \mu < 13.8`

Explanation:

From the question we are told that

The sample mean is
`\= x = 13.6`

The standard deviation is
`\sigma = 1.9`

The sample size is
`n = 189`

given that the confidence level is 85% then the level of significance is mathematically represented as

`\alpha = (100 - 85 )\%`

`\alpha = 0.15`

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

`Z_{(\alpha )/(2) } = 1.44`

Generally the margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) } * (\sigma)/(√(n) )`

=>
`E = 1.44* (1.9)/(√(189) )`

=>
`E = 0.1990`

The 85% confidence interval is mathematically represented as

`\= x - E < \mu <\= x + E`

=>
`13.6- 0.1990 < \mu < 13.6+ 0.1990`

=>
`13.4< \mu < 13.8`