Question

A candy company has calibrated their machines to fill each bag of candy to an average weight of 500 grams with a standard deviation of 5 grams. The company tests the consistency of their machines by taking a sample of 25 bags and weighing them

(5 Pts) What is the probability that the average weight of candy bags in the sample is less than 498 grams?
(5 Pts) What is the probability that the sample average is between 498 and 503?
(5 Pts) Below what value do 90% of the sample means fall?

Answer 1

$Given\;that\, \bar x=500,s=5,n=25\\ (A) P(\bar x<498)=P(Z<\frac{498-500}{\frac{5}{\sqrt{25}}})\\ =P(Z<-2)=0.5-P(0<Z<2)\\ =0.5-0.4772=0.0228\\ (B)P(498<\bar x<503)=P(-2<Z<\frac{503-500}{\frac{5}{\sqrt{25}}})\\ =P(-2<Z<3)\\ =P(0<Z<2)+P(0<Z<3)\\ =0.4772+0.4987=0.9789\\ (C)Let\; Z_{1}=\frac{\bar X-500}{\frac{5}{\sqrt{25}}}=\bar x -500,\\ then,\\ P(Z<Z_{1})=0.90\\ 0.5+P(0<Z<Z_{1})=0.9\\ P(0<Z<Z_{1})=0.4\\ Z_{1}=1.29\\ \therefore 1.29=\bar X-500\\ \bar X=501.29\\$