Name ________________________________________________________ Per ______
Statistics Normal Distributions Worksheet #3 (10 pts.)
I. Basic Concepts Word bank: Bell curve, continuous, less than, mean, median, one, symmetric, zero
1.) Since the normal distribution is _______________________, its probabilities can’t be represented with a
table. Instead, we use a shape with a total area of 1. This special shape is sometimes called the
___________________ .
2.) The normal distribution is symmetric, which means the mean equals the ___________________.
3.) The z-scores of the instances of any normal random variable are also normally distributed, with a mean
of ___________________ and a standard deviation of ___________________.
4.) The table of P(z) values assumes that you’re looking up a ___________________, so if you’re looking up
some other kind of expression, you need to convert it first.
5.) S.A.M stands for ___________________ around the ___________________.
II. Formula matching
6.)
7.) (
_____
)
8.) (
9.)
A.)
_____
B.) (
)
C.) (
)
D.)
(
_____
_____
)
(
)
)
III. Applications
10.) Scores on a test can be approximated with a normal distribution. If the average score was an 86 and
the standard deviation was 3, find the probability that a randomly chosen test has a score…
a.) below 80: _______
b.) between 80 and 90: _______
c.) greater than 90: _______
d.) If 2,000 people took this test, how many can be expected to score greater than a 95? ________
e.) If 2,000 people took this test, around 1,500 could be expected to score between a _________ and a
________ [S.A.M.]
11.) A test is given to government employees. Historically, the scores have been approximately normal with
75 and
6. If an employee scores less than 64 on this test, regulations state that he or she cannot be
hired. Suppose 800 applicants take this test. Estimate the number who will not be hireable: _______
12.) A study found that the mean yearly migration distance of the green turtle was 2400 kilometers and the
standard deviation was 525 kilometers. Additionally, the study found that the migration distances of this
turtle were approximately normal in distribution. Find the probability that a randomly selected green turtle
has a yearly migration distance of…
a.) less than 1800 km: _________
b.) between 1900 and 2500 km :_________
c.) greater than 2200 km: _________
d.) According to this data, approximately 95% of green turtles migrate less than ___________ km per year.
[Round to the nearest km.]
13.) Systolic blood pressure readings in a particular population are approximately normal with a mean of
117 mb. and
5 mb. A researcher wishes to select people for a study but wants to exclude those whose
systolic readings are either in the top 10% and bottom 10%. Therefore, he should only include people
whose readings are between _______ and _______
14.) On a Statistics final exam, the grades were approximately normal in distribution
with a mean of 80 and
4. 650 students took the final. Use this data and the table
to the right to estimate answers to the following questions:
Grade
A
B
C
Fail
Score range
90 or above
[80 – 90)
[70 – 80)
Below a 70
a.) About how many exams received an A grade? ______
b.) About how many exams received a B grade? _________
c.) About how many exams received a passing grade? _______
d.) Now suppose that the Professor wants to “curve” the test and give an “A” grade to the top 50 exams.
What is the lowest score that would qualify for an “A” under this new grading scheme? ________
15.) A designer of motorcycle helmets is marketing a helmet to American men. This designer knows that
the mean American adult male had a head breadth of 6 in, with
1 in, and that the distribution of head
breadths is approximately normal. Find how wide this new helmet would have to be in order to fit…
a.) 75% of all American men: ________
b.) 90% of all American men: _______
c.) 95% of all American men: ________
d.) 99% of all American men: ________
e.) Now suppose this designer eventually settles on a helmet breadth of 7 3/4 in. What percent of American
men will the helmet fit? ___________ [round to the nearest percent.]
f.) Adult American women’s head breadths are also approximately normal, but with a mean of 5.6 in. and
0.8 in. What percent of American women would fit in the 7 3/4-in. helmet? _________
Review Section
16.) In a class of 30 people, 20 play sports and 15 are in the band. 8 very busy students play sports and are
in the band. Suppose a student is chosen at random from this class. Find the probability that he or she…
a.) Plays sports or is in the band: ________
b.) Plays sports, but is not in the band: ________
c.) is in the band, but does not play sports: ________
d.) Is neither in the band nor in sports: ________
e.) Are playing sports and being in the band independent or dependent events? ______________________.
Show the relevant relation (equation or inequality): _______________
17.) How many distinguishable ways are there to arrange 7 red chairs, 5 green chairs and 2 blue chairs if all
chairs of the same color are identical to each other? _____________________
18.) 7% of people in a certain field of study are infected with a certain virus. A test for that virus returns a
positive result 95% of the time that someone has the virus, but it also returns a positive result 10% of the
time someone does not have the virus. What is the probability that a person with a positive result has the
virus? ________
19.) The mean number of Class X solar flares on the sun is 2.4 per year. Assuming that these flares occur
randomly and could occur at any moment in time, find the probability that there are:
a.) exactly 3 Class X flares this year: _______
b.) Less than 3 Class X flares this year: _______
c.) At least 1 Class X flare this year: _______
d.) Exactly 2 Class X flares in the next 6 months: _______
20.) You know that a town with 25,000 households has a mean annual household income of $50,000 per
year with a standard deviation of $3,250. According to Chebychev’s Theorem, at least ______________
households in this town will have a mean annual income between $40,000 and $60,000.
21.) Eleven groups are due to present their projects in History class. Three groups are chosen to present
each day. If the teacher chooses groups to present at random, what are the odds in favor of your group
having to present on the first day? _______
22.) Consider the situation in the previous problem. Let the random variable x = the day on which your
group was chosen to present. For example, if your group was chosen on the first day, x = 1, etc. Find the
mean: ___________
and
standard deviation: ___________ of this variable.
Normal Distributions Worksheet #3 Answer Key
1.) Continuous; bell curve
15a.) 6.67 in.
2.) median
15b.) 7.28 in.
3.) zero; one
15c.) 7.645 in.
4.) less than
15d.) 8.33 in.
5.) symmetric; mean
15e.) 96%
6.) C
15f.) 99.64%
7.) D
16a.) 0.9
8.) B
16b.) 0.4
9.) A
16c.) 0.2333
10a.) 0.0228
16d.) 0.1
10b.) 0.8854
16e.) dependent;
10c.) 0.0918
17.) 72,072
10d.) About 3 (long-term mean of 2.6)
18.) 0.4169
10e.) 82.55 – 89.45
19a.) 0.2090
11.) About 27
19b.) 0.5697
12a.) 0.1271
19c.) 0.9093
12b.) 0.4042
19d.) 0.2169
12c.) 0.6480
20.) 22,360*
12d.) 3,264 km.
21.) 3:8
13.) 110.6 – 123.4 mb.
22.) 2.3636; 1.0679*
14a.) 4
* = your answers may differ slightly due to
rounding.
14b.) 321
14c.) 646
14d.) 85.72
or