Question

Suppose a team of researchers is studying the half-life of a drug in the human body (i.e. how long it takes for 1 2 of the drug to be broken down by the body). They take 50 people, administer a standard dose of the drug, and measure the half-life for each of these people. They find the average half-life to be 7.4 hours. Suppose the variance of half-life is known to be 0.16. Find the 95% confidence interval for population half-life based on this sample. What is the length of this interval?

Answer 1

95% confidence interval for the population half-life based on this sample is [7.29 , 7.51].

Explanation:

We are given that the average half-life to be 7.4 hours. Suppose the variance of half-life is known to be 0.16.

They take 50 people, administer a standard dose of the drug, and measure the half-life for each of these people.

Firstly, the pivotal quantity for 95% confidence interval for the population mean is given by;

P.Q. =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\bar X` = sample average half-life = 7.4 hours

`\sigma` = population standard deviation =
`√(0.16)` = 0.4 hour

n = sample of people = 50

`\mu` = population mean

Here for constructing 95% confidence interval we have used One-sample z test statistics as we know about population standard deviation.

So, 95% confidence interval for the population mean,
`\mu` is ;

P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level

of significance are -1.96 & 1.96}

P(-1.96 <
`(\bar X-\mu)/((\sigma)/(√(n) ) )` < 1.96) = 0.95

P(
`-1.96 * {(\sigma)/(√(n) ) }` <
`{\bar X-\mu}` <

P(
`\bar X-1.96 * {(\sigma)/(√(n) ) }` <
`\mu` <
`\bar X+1.96 * {(\sigma)/(√(n) ) }` ) = 0.95

95% confidence interval for
`\mu` = [
`\bar X-1.96 * {(\sigma)/(√(n) ) }` ,
`\bar X+1.96 * {(\sigma)/(√(n) ) }` ]

= [
`7.4-1.96 * {(0.4)/(√(50) ) }` ,
`7.4+1.96 * {(0.4)/(√(50) ) }` ]

= [7.29 , 7.51]

Therefore, 95% confidence interval for the population half-life based on this sample is [7.29 , 7.51].

The length of this interval is = 7.51 - 7.29 = 0.22