Stat 302 Sample Exam 2 1. Suppose an automated cheese processor cuts cheese in s

Question

Stat 302 Sample Exam 2 1. Suppose an automated cheese processor cuts cheese in such a way that the weight of a pack of cheese is normally distributed. Suppose the mean of this distribution is set to 11.8 ounces, and suppose the standard deviation is 0.15 ounces. (a) If the packages of cheese are labeled to be 12 ounces, what proportion of the packages are underweight (that is, less than 12 ounces)? (b) If only 5% of the cheese packages exceed a certain weight w, what is w? (c) Suppose 5 packages of ABC cheese are chosen at random. What is the probability that at n most one package weighs at least 12 ounces? (d) Suppose two packages are chosen at random. What is the probability that the average weight of the two packages is less than 12 ounces? 2. Suppose the proportion of high school students who work over 20 hours per week during the school year is conjectured to be 25%.

Explanation / Answer

a) proportion of package which are underweight =P(X<12)=P(Z<(12-11.8)/0.15)=P(Z<1.3333)=0.9088

b)for top 5% ; at 95th percentile z score =1.6449

therefore corresponding weight =mean +z*Std deviaiton=11.8+1.6449*0.15=12.05

c)probability that at most one package weight at least 12 ounces =1-P(none exceeds 12 ounce)

=1-(0.9088)5 =1-0.6199 =0.3801

d)for n=2 ; std error of mean =std deviation/(n)1/2 =0.15/(2)1/2 =0.1061

therefore probability that sample mean weight is less then 12 =P(X<12) =P(Z<(12-11.8)/0.1061)=P(Z<1.8856)

=0.9703

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