Math 3118 Exam 5 Solutions

(20 points) 1. Give a list of 10 numbers whose mean is 50 and whose median is 1.

These 10 numbers must sum to 500 for a mean of 50. So just try 1,1,1,1,1,1,1,1,1,491.

(20 points) The birthweight of babies is normally distributed with mean 7.16 lbs, standard deviation of 1.21 lbs.
(a) What percentage of new babies weigh at least 9 lbs?
(b) If your new baby's weight is in the 42nd percentile, what is his/her weight?

Solution: (a) 9-7.16=1.84, which is 1.84/1.21=1.52 standard deviations above the mean. Table 11.6 gives A(1.52)=0.436, so the % above 9 lbs is about 6.4%.
(b) For the 42nd percentile we look up A(z)=.08, so z=0.2, and we move 0.2 standard deviations below the mean. So the answer is 7.16-0.2*1.21=6.92 lbs. about.

(20 points) 3. I roll two fair six sided dice. If I roll a sum of 7, I pay you $3, if I roll a sum of 9, I pay you $2. Otherwise you do not get any money from me. If it costs you $1 to play this game, what are your expected winnings?

Solution: P(roll 7)=6/36,P(9)=4/36, so your expected winnings are 6/36*3+4/36*2-1=-10/36=-27 cents.

(20 points) 4. It is known that 36% of the US population is blue-eyed. Find the approximate probability that in a randomly chosen group of 10,000 people, at least 3650 of them have blue eyes.

Solution: The mean is 10,000*0.36=3600, and the standard deviation is \sqrt(10,000*0.36*0.64)=48. Thus 3650 is 50/48=1.04 standard deviations above the mean. Table 11.6 shows A(1.02)= 0.345 (about) so above this is 0.155, or 15.5%.

(20 points) TRUE or FALSE? (Please give your reason in complete sentences.) In any list of 300 numbers, if the average of the first 100 numbers is 40, and the average of the last 200 numbers is 70, then the average of all 300 numbers is 60.

Solution: TRUE. The first 100 number must sum to 4,000 since their average is 40. The last 200 numbers must sum to 14,000 since their average is 70. So the sum of all 300 numbers is 18,000, and their average is then 18,000/300=60.

Another way to see this is that we take 1/3*40+2/3*70=60, since our first average is from 100 of the 300 numbers, while the second average is from the remaining 200 of the 300 numbers.