Download Math 3A Name: Date: Normal Distribution Quiz Review Key Facts

Math 3A
Name:
Date:
Normal Distribution Quiz Review
Key Facts
outcomes in event
total possible outcomes

Probability =





Mean ( x ) – average of numbers in data set
Deviation ( x - x ) – how far a data value is from the mean
Variance ( s 2 ) – average of squared deviations
Standard Deviation ( s ) – square root of variance
A Bernoulli trial only has two outcomes (success/failure).
æ
ö
o Probability of k successes in n trials = ç n ÷× p k × (1- p)n-k
è k ø
o Mean = np , Variance = np(1- p) , Standard Deviation = np(1- p)



Probability distributions
o Heights of each bar are the probabilities for each outcome
o All the bars’ heights should add to 1
Probability distribution will look more and more like a “bell curve” or normal
distribution as you do more and more trials (Central Limit Theorem)
We can use the normal distribution to answer probability (or percentage)
problems by knowing the percentages between the x ± 1, 2, 3 s values:
-s
-3s
-2s


x +s
+3s
+2s
We can use a graphing calculator to answer probability (or percentage) problems
when the values are not exactly x ± 1, 2, 3 s . The calculator command are found
under 2ND VARS:
o Finding probabilities (or percentages) for a single value
 normalpdf ( value , mean , standard deviation )
o Finding probabilities (or percentages) between two values
 normalcdf ( lower value , upper value , mean , standard deviation )
o Finding probabilities (or percentages) greater than a value
 normalcdf ( lower value , really big number , mean , st. deviation )
o Finding probabilities (or percentages) less than a value
 normalcdf ( really small number , upper value , mean , st. deviation )
A z-score can help compare results between different (but similar) events. To
calculate: z-score = (value of interest – mean)/ standard deviation
Practice
1. Fill in the following table by finding the probabilities for the number of heads when
flipping a fair coin 3 times. Then, use the table to create a probability distribution.
# of heads
Probability
0
1
2
3
2. Which of the following probability distributions are
impossible?
A.
B.
C.
D.
3. A decathlon is a combined event consisting of ten track and field events. Here are
results for one competitor in three of those events, including the mean and standard
deviation for all the competitors.
100-yard dash (sec) Shot put (feet)
Long jump (feet)
10.0
60
26
Mean
0.2
3
0.5
Standard Dev.
10.3
64
26.25
Competitor A
a. Calculate how many standard deviations from the mean Competitor A was in the
three events (calculate z-scores).
b. Which is Competitor A’s best event? Use the z-scores to decide. Hint: Think
carefully about each event.
4. Thumbtacks have a 60% chance of landing point up. You toss 150 thumbtacks.
a. Find the mean, variance and standard deviation for the number of thumbtacks that
land point up.
x=
s2 =
s=
b. You got 105 thumbtacks to land point up. How many standard deviations away
from the mean is this result? (Calculate the z-score)
c. Is it unusual to get 105 thumbtacks to point up? Use the z-score to decide.
5. What is the probability of getting 10 questions correct on a 30 questions multiplechoice test where each question has 4 options?
6. At a certain high school, 35% of the students participate in music. Find the mean,
variance, and standard deviation for the number of students who will say they
participate in music if you ask 100 students.
7. Recall the statistics for a multiple-choice question with 5 options:
x = 0.2
a.
s 2 = 0.16
s = 0.4
Find the mean, variance, and standard deviation for the number of questions
correct on a 25-question test.
b. Find the mean, variance, and standard deviation for the number of questions
correct on a 50-question test.
Directions for the problems 8 - 11:
 Label and shade in the normal distribution appropriately
 Decide if the ranges asked about are exactly x ± 1, 2, 3 s . Then either:
o Use the normal distribution picture on the first page to answer
o Write out the normalpdf or normalcdf command needed and use your
calculator to answer
8. The weights of adult cats are normally distributed with a mean of 14 pounds and a
standard deviation of 3 pounds.
a. Approximately what percent of cats have a weight between 11 and 17 pounds?
b. Approximately what percent of cats have a weight between 14 and 20 pounds?
c. Give a range in which approximately 99.7% of all cat’s weights should lie.
9. Use the normal distribution to approximate the number of sixes when rolling 600
number cubes. Find the probability that the number of sixes recorded is 80 or fewer.
Hint: You must calculate the mean and st. deviation
first.
10. Bags of concrete mix produced by a certain manufacturer weigh on average 90 lb.
The weights are normally distributed with a standard deviation of 0.4 lb.
a. What percentage of the bags weigh more than the average?
b. What percentage weigh less than 89 lb?
c. If the lightest 2.5% of bags are to be repacked, what is the cutoff weight to determine
which bags are chosen?
11. Assume the heights of men are normally distributed with a mean of 69.5 in. and a
standard deviation of 2.9 in.
a. What percentage of men are 70 inches tall?
b. What percentage of men are between 63.7 and 78.2 inches tall?
c. What percentage of men are taller than the average woman who is 65.5 inches?
d. You are starting a men's clothing store and to minimize startup costs you will only
stock suits for the middle 95% of men. Find the minimum and maximum heights
of men for which suits will be stocked.
12.
What is the probability of a “300 hitter” getting 10 hits in 20 at-bats?
13. What is the probability of getting 5 odd numbers if you roll 10 number cubes?
14. Find the mean, variance, and standard deviation for the number of snowy days in the
next week if there is a 40% chance of snow each day.
15. A campaign manager thinks 45% of voters prefer his candidate. Find the mean,
variance, and standard deviation for the number of voters who will say they prefer
this candidate if you ask 500 voters.
Answers
1.
# of heads
0
1
2
3
Probability
0.125
0.375
0.375
0.125
2. A (adds to less than 1)
C( adds to more than 1)
10.3-10.0
64 - 60
26.25 - 26
3. a. dash:
jump:
=1.5 shot put:
=1.33
= 0.5
0.2
3
0.5
b. Their best event is the shot put (even though they were further above the mean in
the dash, it is desirable to be below the mean in running!)
4. a. x = (150)(0.6) = 90 s 2 = (150)(0.6)(1- 0.6) = 36 s = 36 = 6
105 - 90
b.
= 2.5 st. deviations above mean
6
c. Unusual because it is more than 2 st. deviations away from the mean
æ
ö
5. ç 30 ÷× (0.25)10 × (1- 0.25)30-10 = 0.091
è 10 ø
2
6. x =100(0.35) = 35 s =100(0.35)(1- 0.35) = 22.75
7. a. x = 0.2 × 25 = 5
s 2 = 0.16 × 25 = 4
b. x = 0.2 × 50 =10
s 2 = 0.16 × 50 = 8
s = 22.75 = 4.770
s = 4 =2
s = 8 = 2.828
8. a. 34+34=68%
b. 34+13.5=47.5%
c. within 3 st. deviations – 5 to 23 pounds
9. normalcdf ( 0, 80, 100, 9.13 ) = 0.014 or 1.4%
10. a. 50%
b. normalcdf ( 0, 89, 90, 0.4 ) = 0.0062 or 0.62%
c. more 2 st. deviation below mean – 89.2 pounds
11. a. normalpdf ( 70, 69.5, 2.9 ) = 0.136 or 13.6%
b. 13.5+34+34+13.5+2.35 = 97.35%
c. normalcdf ( 65.5, 1000, 69.5, 2.9 ) = 0.916 or 91.6%
d. within 2 st. deviations – 63.7 to 75.3 inches
æ
ö
12. ç 20 ÷× (0.3)10 × (1- 0.3)20-10 = 0.0308
è 10 ø
æ 10 ö
÷× (0.5)5 × (1- 0.5)10-5 = 0.246
13. ç
è 5 ø
14. x = 7(0.4) = 2.8
s 2 = 7(0.4)(1- 0.4) =1.68
15. x = 500(0.45) = 225
s = 1.68 =1.296
s 2 = 500(0.45)(1- 0.45) =123.75
s = 123.75 =11.124