Question

16. In a study of the effect of prenatal cocaine use on infants, the following sample data were obtained for weights at birth: n = 101, x 2700 grams  , and s = 645 grams (based on data from “Cognitive Outcomes of Preschool Children with Prenatal Cocaine Exposure,” by Singer et al., Journal of the American Medical Association, Vol. 291, No. 20). Use the sample data to construct a 95% confidence interval estimate of the standard deviation of all birth weights of infants born to mothers who use cocaine during pregnancy. Round to the nearest whole gram.

Answer 1

First find degrees of freedom:

Degrees of Freedom = n - 1

Degrees of Freedom = 101 - 1

Degrees of Freedom = 100

Calculate α: α = 1 - confidence%

α = 1 - 0.95

α = 0.05

Find low end confidence interval value:

αlow end = α/2

αlow end = 0.05/2

αlow end = 0.025

Find low end χ^2 value for 0.025

χ^2(0.025) = 129.5613 <--- Value can be found on Excel using =CHIINV(0.025,100)

Calculate low end confidence interval total:

Low End = Square Root((n - 1)s^2/χ^2α/2)

Low End = √(100)(416025)/129.5613)

Low End = √41602500/129.5613

Low End = √321102.82931709

Low End = 566.6594

Find high end confidence interval value:

αhigh end = 1 - α/2

αhigh end = 1 - 0.05/2

αhigh end = 0.975

Find high end χ2 value for 0.975 χ^2(0.975) = 74.2219 <--- Value can be found on Excel using =CHIINV(0.975,100)

Calculate high end confidence interval total:

High End = Square Root((n - 1)s^2/χ^21 - α/2)

High End = √(100)(416025)/74.2219)

High End = √41602500/74.2219

High End = √560515.15792509

High End = 748.6756

our 95% confidence interval is : 567 < σ < 749 <---- This is our 95% confidence interval estimate of the standard deviation of all birth weights of infants born to mothers who use cocaine during pregnancy.