Question

1) A brewery's filling machine is adjusted to fill bottles with a mean of 32.4 oz. of ale and a variance of 0.003. Periodically, a bottle is checked and the amount of ale noted.

(a) Assuming the amount of fill is normally distributed, what is the probability that the next randomly checked bottle contains more than 32.49 oz? (Give your answer correct to four decimal places.)

(b) Let's say you buy 98 bottles of this ale for a party. How many bottles would you expect to find containing more than 32.49 oz. of ale? (Round your answer up to the nearest whole number.)

bottles

You may need to use the appropriate table in Appendix B to answer this question.

2) The weights of ripe watermelons grown at Mr. Smith's farm are normally distributed with a standard deviation of 2 lb. Find the mean weight of Mr. Smith's ripe watermelons if only 2% weigh less than 16 lb. (Give your answer correct to one decimal place.) (with step explanations please)

Answer 1

a)

X ~ N ( µ = 32.4 , σ = 0.0548 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 32.49 ) = P(Z > (32.49 - 32.4 ) / 0.0548 )
= P ( Z > 1.64 )
= 1 - P ( Z < 1.64 )
= 1 - 0.9495
= 0.0505

b)

We expect, 0.0505 * 98 = 5 bottles containing more than 32.49 oz. of ale.

2)

P(X < 16) = 0.02

P(Z < ( x - ) / ) = 0.02

From Z table, z-score for the probability of 0.02 is -2.0537

( x - ) / ) = -2.0537

( 16 - ) / 2 = -2.0537

= 20.1