Question

Corn soy blend (CSB) is often provided to relief agencies for distribution in refugee camps or as emergency relief. Production of CSB is carefully monitored to ensure that the product has appropriate levels of certain vitamins and minerals for good nutrition. 8 bags of CSB were randomly taken from a large shipment and the amount of vitamin C was measured. The measured concentration in milligrams of vitamin C per 100 grams of CSB were

26, 31, 23, 22, 11, 22, 14, 31

a. Construct a 95% confidence interval for the true mean amount of Vitamin C per 100 grams of CSB.

b. Construct a 95% upper bound for the true variance of vitamin C per 100 grams of CSB.

c. Construct a 95% prediction interval of vitamin C per 100 grams of CSB.

Answer 1

a)
M = 22.5 and degrees of freedom =n-1= 8-1= 7
t critical value = 2.36
sM = √(7.192/8) = 2.54

μ = M ± t(sM)
μ = 22.5 ± 2.36*2.54
μ = 22.5 ± 6.011

M = 22.5, 95% CI [16.489, 28.511].

You can be 95% confident that the population mean (μ) falls between 16.489 and 28.511.

b) Confidence Interval Formula for σ^2 is as follows: (n - 1)s^2/χ^2(α/2) < σ^2 < (n - 1)s^2/χ^2(1 - α/2) where: (n - 1) = Degrees of Freedom,

s2 = sample variance

and α = 1 - Confidence Percentage

First find degrees of freedom: Degrees of Freedom = n - 1

Degrees of Freedom = 8 - 1

Degrees of Freedom = 7

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) = 16.0128 <--- Value can be found on Excel using =CHIINV(0.025,7)

Calculate low end confidence interval total:

Low End = (n - 1)s^2/χ2α/2

Low End = (7)(51.7)/16.0128

Low End = 361.9/16.0128

Low End = 22.6007

Find high end confidence interval value:

αhigh end = 1 - α/2

αhigh end = 1 - 0.05/2

αhigh end = 0.975

Find high end χ^2 value for 0.975 χ^2(0.975) = 1.6899 <--- Value can be found on Excel using =CHIINV(0.975,7)

Calculate high end confidence interval total: High End = (n - 1)s^2/χ^2(1 - α/2)

High End = (7)(51.7)/1.6899

High End = 361.9/1.6899

High End = 214.1547

our interval answer: 22.6007 < σ^2 < 214.1547 <---- This is our 95% confidence interval

a 95% upper bound for the true variance of vitamin C per 100 grams of CSB is 214.1547

Note: I have done the first two questions step by step. Please repost question C with data on question board. Thank you.