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### 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 Today with access to the Internet, the old way of solving these problems is obsolete, though a student who learns to solve the the old-fashioned way may be able to understand the subject better.)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?

You can calculate a probability from a normal table on the Internet--there 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:

Go to one and use it to solve these:

Suppose that a gym teacher fins that the time it takes students to run a 100-yard 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 working-backward 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?

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