Problems: The Normal Curve
(These questions were written assuming that students had
access to a table with Normal curve probabilities. Today
with access to the Internet, the old way of solving these
problems is obsolete, though a student who learns to solve
the the oldfashioned way may be able to understand the
subject better.)
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?
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?
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?
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?
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?
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?
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.
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?
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?
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?
Answers here.
You can calculate a probability from a normal table on the
Internetthere are quite a few sites with Normal curve
tables and calculators. Here are several that allow you to
enter numbers and it gave you the result:
http://davidmlane.com/hyperstat/z_table.html
http://math2.org/math/stat/distributions/zdist.htm
http://25yearsofprogramming.com/javascript/probability.htm
Go to one and use it to solve these:
Suppose that a gym teacher fins that the time it takes
students to run a 100yard dash is normally distributed with
a mean of 13.2 seconds and a standard deviation of .6
seconds.
 a) What percentage of students will be able to run
100 yards in 11 seconds or less?
 b) What percentage of students will take from 12 to
14 seconds?
2. Suppose the standard deviation is 1.1 seconds. Rework
the probabilities. ___ ____
Here is another normal curve calculator that should allow
you to answer the workingbackward problems:
http://davidmlane.com/hyperstat/z_table.html
Use it to answer the following:
c) What are the middle limits in which 50% of the
students will be?
d) How slow must a student be to be in the slowest 5% of the
students?
More Problems:
11. A grenade manufacturer has a product with a fuse time
that averages five second with a standard deviation of .25
seconds. Assume that the fuse time is normally
distributed.
 a) What is the probability of the grenade exploding
in less than four seconds?
 b) What is the probability that the explosion will
occur between 4.9 and 5.1 seconds?
12. Assume that the average height of a large group of
adult males is distributed normally with a mean of 70 inches
and standard deviation of three inches.
 a) What percentage of this population would be
between 70 and 74 inches tall?
 b) What percentage would be less than 69 inches?
 c) What percentage would between 68 and 72
inches?
 d) What is the probability that if two persons are
selected at random from this population, both will be
over 75 inches tall?
13. The number of weeds per square foot in my garden is
approximately normally distributed with a mean of 19 and a
standard deviation of 4.
 a) What percentage of square feet have less than 15
weeds in them?
 b) What percentage have more than 17?
 c) What percentage have between 18 and 21?
