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### Answers: The Normal Curve

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1. Suppose SAT scores are normally distributed with a mean of 1000 and a standard deviation of 100.

a) What percentage will be between 900 and 1100?
b) What percentage of scores will be between 1100 and 1200?
c) What percentage of scores will be below 850?
d) How high must the score be to be in the top 2% of all scores?

a) 68.27; b) 13.59; c) 6.68%
d) z = 2.054 = (x-1000)/100; x = 1000 + 205 = 1205.

2. Suppose that the length of time people can hold their breath is normally distributed with a mean of 80 seconds and a standard deviation of 12 seconds.

a) What percentage of people will be able to hold their breath more than 80 seconds?
b) What percentage of people will be able to hold their breath between 60 and 80 seconds?
d) What percentage of people will be able to hold their breath more than 110 seconds?
e) What percentage of people will be able to hold their breath between 90 and 100 seconds?
f. How long would you have to be able to hold your breath to be in the top 5 percent of this population?

a) 50%; b) 45.22%; c) 0.6%; d) 15.45%
f) z = 1.545 = (x-80)/12; x = 80 + 18.5 = 98.5

3. Suppose a large orchard produces apples that have a mean weight of 112 grams with a standard deviation of 8 grams. (Please draw a picture showing what you are looking for.)

a) What percentage of apples will weigh less than 120 grams?
P(x < 120)
b) What percentage of apples will weigh between 112 and 120 grams?
P(112 < x < 120) =
c) What is the probability that an apple chosen at random will weigh between 120 and 132 grams?
P(120 < x < 132) =
d) What percentage will weigh between 100 and 120 grams?
P(100 < x < 120) =
e) How small must an apple be to be in the smallest 10% of apples?
f) What range of apple sizes will make up the middle 50% of the population?

a) 84.13%; b) 43.13%; c) 15.24%; d) 77.45%
e) z = -1.28 = (x-112)/8; x = 112 + 10.24 = 122.24.
f) z = -.6745 = (x-112)/8; x = 112 - 5.4 = 106.6 for the lower number; z = .6745 = (x-112)/8; x = 112 + 5.4 = 117.4 for the upper number.

4. Assume that a machine can fill cereal boxes with a mean of 36 ounces and a standard deviation of 1 ounce.

a) What percentage of boxes will have between 37 and 38 ounces?
b) What percentage of boxes will have less than 35 ounces?
c) How light must a box be to be in the lowest five percent of all boxes?

a) 13.59%; b) 15.87%
c) z = -1.64 = (x - 36)/1; x = 36 -1.64 = 34.36.

5. Suppose a variety of bananas produces bunches that have a mean weight of 70 pounds with a standard deviation of 3 pounds. (Please draw a picture showing what you are looking for.)

a) What is the probability that a bunch will weigh between 70 and 74 pounds?
P(70 < x < 74) =
b) What is the probability that a bunch chosen at random will weigh between 71 and 75 pounds?
P(71 < x < 75) =
c) What is the probability that a bunch will weigh between 68 and 76 pounds?
P(68 < x < 76) =
d) What is the probability that a bunch will weigh less than 63 pounds?
P(x < 63) =
e) How small must a bunch be to be in the smallest 12% of bunches?

a) .4087; b) .3217; c) .7248; d) .0098
e) z = -1.175 = (x - 70)/3; x = 70 - 3.5 = 76.5

6. Suppose that the life of a particular brand of watch battery is 1000 days on average with a standard deviation of 50. Suppose further that the lifetimes of these batteries are normally distributed.

a) What percentage of these batteries will last more than 920 days?
b) How long must a battery last so that it is in the top 10% of all batteries?

a) 94.52
b) z = 1.28 = (x - 1000)/50; x = 1000 + 64 = 1064.

7. IQ scores are supposed to be normally distributed with a mean of 100 and a standard deviation of 15.

a) If a person scores 120 on an IQ test, what percentage of the population is above him or her?
b) How low would a person have to score to be in the lowest 1% of the population.

a) 9.12%
b) z = -2.33 = (x - 100)/15; x = 100 + 15*(-2.33) = 65.

8. Suppose that the amount of time (in minutes) need for students to complete a test is N(40,10)

a) What percentage of students will finish in less than 50 minutes?
b) What percentage will finish in less than 25 minutes?
c) What is the probability that a student selected at random will take between 35 and 45 minutes?
d) How many minutes will elapse before 90% of the students are finished?

a) 84.13; b) 6.68% c) 38.92%
d) z = 1.28 = (x - 40)/10; x = 40 + 12.8 = 52.8

9. A bakery sells and average of 1200 donuts per day with a standard deviation of 200 donuts.

a) If it bakes 1300 donuts, what is the probability that it will sell all of them before the end of the day. (Assume a Normal distribution.)
b) Assuming a Normal distribution, how many donuts must it bake to have a probability of .80 that it will not run out?

a) .3085
b) z = .8416 = (x - 1200)/200; x = 1200 + 168 = 1368

10. Suppose a variable is distributed normally with a mean of 15 and a standard deviation of 3.

a) What is the probability that x<13?
b) What is the probability that 14<x<17?
c) What is the probability that 13<x<14?
d) What values include the middle 40% of the distribution?
e) If two items are drawn from this distribution, what is the probability that at least one will have a value of 12 or less?

a) .2525; b) .3781; c) 11.69;
d) lower z = -.5244 = (x - 15)/3; x = 15 - 1.6 = 13.4; upper z = .5244 = (x - 15)/3; x = 15 + 1.6 = 16.6;
e) The probability that at least one will have a value of 12 or less is 1 - probability that neither will have a value of 12 or less. Assuming that they are independent, the probability of both being 12 or greater can be found by finding the probability that one draw will be twelve or greater and multiplying it by itself. Twelve is one standard error below the mean, the probability of being one standard error below the mean or more is .1587 and the probability of being 12 or above is .8414. Therefore the probability of both draws being above 12 is .84.14*.8414 = .7079. The probability of at least on draw being below 12 is 1 - .7079 = .2921.

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