Improve Your Understanding of College Level Statistics Problems

Find the length of the confidence interval given the following data 

S=3 n=275 confidence level 95 %

Brämhults Juice, based in Sweden, produces chilled fruit and vegetable juices. A manager at Brämhults is considering purchasing a new machine to bottle 16-fluid-ounce (473-milliliter) bottles of 100% pure carrot juice and wants an estimate of the differ- ence in the mean filling weights between the new machine and the old machine. Random samples of bottles of carrot juice that had been filled by both machines were obtained. Estimate the difference in the mean filling weights between the new and the old machines? Discuss the assumptions. Use a = 0.10.

Assume that we want to construct a confidence interval. Do one of the​ following, as​ appropriate: (a) find the critical value

tα/2​,

​(b) find the critical value

zα/2​,

or​ (c) state that neither the normal distribution nor the t distribution applies.

Here are summary statistics for randomly selected weights of newborn​ girls:

n=162​,

x=30.1

​hg,

s=6.1

hg. The confidence level is

99​%.

Deriving the confidence interval
${\displaystyle P(-{\mathrm{\Phi }}_{\alpha }^{-1}<X<{\mathrm{\Phi }}_{\alpha }^{-1})=1-2\alpha }$
Let ${\displaystyle P\sim \mathcal{N}(0,1)}$
$\begin{array}{rlrlr} & P(-{\mathrm{\Phi }}_{\alpha }^{-1}<X<{\mathrm{\Phi }}_{\alpha }^{-1})& & & \left(1\right)\\ & P(X<{\mathrm{\Phi }}_{\alpha }^{-1})-P(X<-{\mathrm{\Phi }}_{\alpha }^{-1})& & P(A<X<B)=P(X<B)-P(X<A)& \left(2\right)\\ & \mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right))-\mathrm{\Phi }(-{\mathrm{\Phi }}_{\alpha }^{-1}\left)\right)& & P(X<A)=\mathrm{\Phi }\left(A\right)& \left(3\right)\\ & \mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right))-(1-\mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right)\left)\right)& & \mathrm{\Phi }(-x)=1-\mathrm{\Phi }\left(x\right)& \left(4\right)\\ & \alpha -1+\alpha & & f\left(f^{-1}\right)=id& \left(5\right)\\ & -1\times (1-2\alpha )& & & \left(6\right)\end{array}$