Answer to Question #150742 in Statistics and Probability for abed

Question

A candy company has calibrated their machines to fill each bag of candy to an average weight of 505 grams with a standard deviation of 4 grams. The company tests the consistency of their machines by taking a sample of 25 bags and weighing them.



a- What is the probability that the average weight of candy bags in the sample is less than 503 grams?

b- What is the probability that the sample average is between 503 and 506?

c- Below what value do 95% of the sample means fall?

Answer 1

$\mu=505$

$\sigma=4$

$n=25$

Let's denote $\bar x$ sample mean

a) $P(\bar x<503)=P(Z<\frac{503-\mu}{\frac{\sigma}{\sqrt n}})=P(Z<\frac{503-505}{\frac{4}{\sqrt 25}})=P(Z<-2.5)=0.0062$

b) $P(503<\bar x<506)=P(\frac{503-505}{\frac{4}{\sqrt 25}}<Z<\frac{506-505}{\frac{4}{\sqrt 25}})=P(-2.5<Z<1.25)=$

$=P(Z<1.25)-P(Z<-2.5)=0.8944-0.0062=0.8882$

c) $P(\bar x<k)=95\%=0.95 \implies P(Z<1.645)=0.95$

$\frac{k-505}{\frac{4}{\sqrt 25}}=1.645 \implies k=506.316$

Answer:

a) 0.0062

b) 0.8882

c) 506.316