Download Unit 3 Student Study Guide and Answers

Unit 3 Study Guide
Important Concepts and Vocabulary
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Random Variable: a rule that assigns a number to each outcome of an experiment.
○ Finite Discrete: Has a finite number of values.
○ Infinitely Discrete: Has infinitely many values.
○ Continuous: Has values which are in intervals of real numbers.
Expected Value: associating values with their probabilities.
E= Value x Probability
Variance:a measure of the spread of data. The larger the variance the larger the spread.
Standard Deviation: measure of the spread of data using the same units as the data.
○ To Find SD and Var in the calculator:
■ Type X values into L1 and L2
■ STAT, CALC, 1-Var Stats, ENTER
Standard Deviation= √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 and Variance= 𝑆𝐷 2
Properties of Binomial Experiment
○ Number of trials is fixed.
○ Only outcomes are Success or Failure.
○ Probability of Success is equal in each trial.
○ Trials are independent.
Variables of a Binomial Experiment
○ n= number of trials
p= probability of success
○ x= number of successes
q= probability of failure
Binomial Experiment in the Calculator
○ 2nd, VARS, BinomPDF (n,p,x)
Odds- 𝑬 ÷ 𝑬𝒄 (in favor) 𝑬𝒄 ÷ 𝑬 (against)
Z-scores: the number of standard deviations a value is from the mean𝑉𝑎𝑙𝑢𝑒 − 𝑀𝑒𝑎𝑛 ÷
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 OR
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𝑥−𝜇
𝜎
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Empirical Rule: nearly all values fall within 3 standard deviations of the mean in normal
distribution
Normal Distribution Curve
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Applications of Normal Distribution:
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To find probabilities that are not perfect standard deviations from the mean, use
normalCDF.
■ In the calculator: 2nd, VARS, normalcdf (lower, upper, mean, standard
deviation)
○ To find percentiles, work backwards by using INVnorm
■ In the calculator: 2nd, VARS, invnorm (area to left, mean, standard
deviation)
Simulations: model random events to match real outcomes
○ In the calculator: 5, STO, MATH, PRB, rand, ENTER. To ensure you get the
same results.
Once you determine the conditions of the situation, enter your data into the calculator to
conduct your simulation.
Practice Questions
1. On a normal distribution curve what percentage of scores are:
a. Below the mean?
b. Above the mean?
c. Within one standard deviation of the mean?
d. Within two standard deviations of the mean?
e. Within three standard deviations of the mean?
2. The average salary for a professional football player is $ 69,000 with a standard
deviation of $4,250. What is the probability that the average football player makes
between $60,000 and $75,000 per year?
3. A fair die is rolled six times. Calculate the probability of obtaining exactly three 6s with
binomial cdf.
4. Your wallet holds 4 different types of coins (pennies, nickels, dimes, and quarters).
Design a simulation as if you had equal amounts of each coin and you pulled out a eight
coins at a time. Write it out in an equation, DO NOT CONDUCT.
For questions 1-2 use the following information: Phone prices are normally distributed. The
average price of a phone is $500 with a standard deviation of $50.
5. What percent of computers cost less than $400?
6. What is the most expensive 10% of phones cost more than the least amount?
7. The mean of life of a TV is 25,000 hours. The standard deviation is 2,500 hours. If Best
Buy purchases 6000 TVs, how many TVs would you expect to last more than 28,000
hours?
8. The mean score on the SAT (math & verbal) is 1500, with a standard deviation of 240.
The ACT, another college entrance exam, has a mean score of 21 with a standard
deviation of 6.
a. If Kathy scored 1740 on the SAT, calculate his z-score.
b. If Bobby scored 30 on the ACT, calculate her z-score.
c. Who performed better on his or her admissions test compared to his or her
peers? Explain.
9. The height of women aged 20 to 29 are approximately normal with mean 64 inches and
standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with
standard deviation of 2.8 inches. John and his sister June both play basketball for N.C.
State University. John is 81 inches tall; June is 74 inches tall.
a.) Compared to their respective peers, who is the tallest?
b.) How tall would a woman be who has a z-score of 1.5?
c.)If a man has a z-score of -0.5 and a woman has a z-score of 1.2, which is taller?
10. Sketch a normal curve and solve.
The percentage impurity of a chemical can be modeled by a normal distribution with a
mean of 5.8 and a standard deviation of 0.5. Obtain the probability that a sample of the
chemical has percentage impurity between 5 and 6.
11. There is a 25% chance that MCHS will lose the lacrosse game. What are the odds that
they lose?
12. Brandy and her friend Jake are both enrolled in Math 3, but at different schools. On the
Unit 1 test, Brandy scored 90. Her class average was 86 with a standard deviation of
2.6. Jake scored 88. His class average was a 79 with a standard deviation of 2.2. Which
student did better in respect to his/her class? Explain.
13. Three marbles are selected at random without replacement from a box containing eight
green marbles and five purple marbles. Let the random variable X denote the number of
green marbles drawn.
A. List the outcomes of the experiment and find the value assigned to each outcome of the
experiment by the random variable X.
outcomes
Value of X
B. Find P(X=3). Show work!
Answer Key
1.
A .50% B. 50% C. 68% D. 95% E.97.5%
2. .904 normcdf
3. .054
(6𝑛𝐶𝑟3) ∗ (1/6)3 ∗ (5/6)6−3or 𝑏𝑖𝑛𝑜𝑚𝑝𝑑𝑓(6,1/6,3)
4. STO→5
0-26: pennies
27-50: nickels
51-75: dimes
76-100:quarters
RandInt(1,100,8)
5. .023 normcdf
6. 436 Invnorm
7. 690 Tires normcdf
8. a.) z=(1740-1500)/(240) z=1 b.) z=(30-21)/(6) z= ~1.5 c.) Bobby did better than
Kathy because she had a higher z-score.
9. a.) z(w)=(74-64)/(2.7)=3.7
z(m)=(81-69.3)/(2.8)=4.18 -> John b.) x=68.05 c.) The
woman because her z-score is higher.
10. normalcdf (5, 6, 5.8, 5)= ~6.01
11. .25/.75=25/75=25:75=5:15=1:3
12. Brandy’s-
𝟗𝟎−𝟖𝟓
=1.9
𝟐.𝟔
Jake’s-
88−79
=4.1
2.2
Jake scored better because his z-score was
higher than Brandy’s.
13. A.
outcom
es
RRR
RRW
RWR
RWW
WWW
WWR
WRW
WRR
Value
of x
3
2
2
1
0
1
1
2
B. RRR ⅛=0.125