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TEST 1 BEGINS .MAKE SURE YOU HAVE REGISTERED AND RECORDED YOUR TIME.
IF YOU ARE LATE, THEY WILL NOT ADMIT YOU.
Math 2311
Sections 4.1, 4.2 and Review for Test 1 (If you have more questions)
4.1 - Density Curves
A density curve is a graph whose area between it and the x-axis is equal to one. These
graphs come is a variety of shapes but the most familiar “normal” graph is bell shaped.
The area under the curve in a range of values indicates the proportion of values in that
range.
Example: Thing about a density curve that consists of two line segments. The first goes
from the point (0, 1) to the point (.4, 1). The second goes from (.4, 1) to (.8, 2) in the xy
plane.
Sketch:
What percent of observations fall below .4?
What percent of observations lie between .4 and .8?
What percent of observations are equal to .4?
Example: Consider a uniform density curve defined from x = 0 to x = 6.
Sketch:
What percent of observations fall below 2?
What percent of observations lie between 2 and 3?
Find the median.
Skewness and curves:
Bell Shaped (normal)
Skewed Right
Skewed Left
4.2 - The Normal Distribution
A density curve that is symmetric, single peaked and bell shaped is called a normal
distribution.
The normal distribution with mean µ and standard deviation σ is represented by N(µ, σ).
The Empirical Rule:
The Empirical Rule states if a distribution has a normal distribution,
1. Approximately 68% of all observations fall within one standard deviation of the
mean.
2. Approximately 95% of all observations fall within two standard deviations of the
mean.
3. Approximately 99.7% of all observations fall within three standard deviations of the
mean.
Example:
The length of time needed to complete a certain test is normally distributed with mean 60
minutes and standard deviation 10 minutes.
What is the probability that someone will take between 40 and 80 minutes to complete the
test? Sketch the distribution and shade in the area in question.
Find the interval that contains the middle 68% of completion times for all people taking
the test.
What percent of people take more than 80 minutes to complete the test?
What if our values are not exactly within one, two or three standard deviations from the
mean? Probabilities for these can still be found a number of ways, one of which we will
explore in the next section. Using R, the probability can be found with the command
pnorm(X, µ, σ). Note that the command in R only gives the probability that X is less than a
given value. If we need to find the probability that X is greater than the given value, we
will need to subtract the answer from 1.
With the TI-83 and TI-84 calculator, the command is
normalcdf(lower_limit, upper_limit, µ, σ).
Continuing the example above,
What is the probability that someone will take less than 45 minutes to complete the test?
What is the probability that someone will take more than 30 minutes to complete the test?
How long would it take someone to finish the test if they are in the top10% of the times?
Popper 05
1 If a group of students have test scores that are normally distributed with a mean of 82
and a standard deviation of 4, half of the students made below a grade of:
a. 78
b. 74
c. 82
d. 70
e. none of these
TEST 1 BEGINS THURSDAY 2/13!!!! MAKE SURE YOU HAVE REGISTERED AND
RECORDED YOUR TIME. IF YOU ARE LATE, THEY WILL NOT ADMIT YOU.
Math 2311
REVIEW FOR TEST 1
Review for Test 1
7 t/f (4 pts each) and m/c (8 pts each)
3 f/r (either 14 or 16 pts each)
Sections 1.1 – 3.3
Review sheet posted on Homework page
Practice Test under online assignments on CASA
FORMULAS (not necessarily in this order):
å (xi - x )2
n
Pn = n(n -1)(n - 2)...3× 2 ×1 = n!
n -1
n
Pr =
n
s2 =
i=1
s = s2
P=
 A  B  c  A c B c
(
s = Var[X] = ( x1 - m X ) p1 + ( x2 - m X ) p2 +
2
2
X
= å ( xi - m X ) pi
2
E[X + Y ] = E[ X] + E[Y ]
= Var[X + Y ] = Var[X] + Var[Y ]
E[X - Y ] = E[ X - Y ] = E[X] - E[Y ]
2
X+Y
s 2X-Y = Var[X - Y ] = Var[X] + Var[Y ]
 2  np 1  p 
1
E[X] = m =
p
1- p
s2 = 2
p
( ) ( ) (
mX = E[X] = x1 p1 + x2 p2 +
P(E Ç F)
P(F)
m = E éë X ùû = np
)
P EÈF = P E + P F - P EÇF
P(E c ) = 1 - P(E)
s
n!
r!s!t!
æ n ö
n!
C
=
ç
÷ = r!(n - r)!
n r
è r ø
n(E)
P(E) =
n(S)
P(E | F) =
n!
(n - r)!
2
+ ( xn - m X )
2
)
+ xn pn
2
2
s
=
Var[X]
=
E[X
] - ( E[X])
X
pn
2
E[W ] = E[aX + b] = aE[X] + b
s W2 = Var[W ] = Var[aX + b] = a 2Var[X]
n
P( X  k )    p k (1  p)n k
k 
P( X ³ k) = 1- P( X £ k -1 )
(
P( X  n)  (1  p) n1 p
P( X  n)  (1  p) n
)
What is the difference between a sample and a population?
What is the difference between continuous and discrete random variables?
Measures of center?
Measures of spread?
What does it mean to be “resistant to outliers?”
How do you find outlier boundaries?
Counting techniques –
What is the difference between combinations and permuatations?
Sets (intersections, unions, complements)
Probability – What formula to use??
Some extra examples:
Given:
U = {1,2, 3, 4,5,6, 7,8,9,10}
A = {2, 4,6,8,10}
B = {1,2, 3, 4}
C = {4,5,6,9}
Find:
(B È C)C =
A Ç (B È C)C =
(AC Ç C)C =
Suppose P(A) = 0.25 and P(B) = 0.30. If events A and B are independent then,
P(A Ç B) =
P(A È B) =
P(A | B) =
A distribution of grades in an introductory statistics class (where A = 4, B = 3, etc) is:
X
0 1 2 3 4
P(X) .10 .15 .35 ?? .05
Find P(X = 3)
Find
P(1 £ X < 3)
E[X] =
VAR[X] =
P ( X  X 0 )  0.5
Find the lowest grade X 0 such that
Have questions for me to work from the review sheet or practice test!!!!
Popper 05
2 The test scores of a class of 20 students have a mean grade of 71.6 and the test scores
of another class with 14 students has a mean grade of 78.4. What is the mean of the
combined group?
a. 74.4
b. 75
c. 71.6
d. 78.4
e. none of these
Popper 05
3 If a constant value is added to all data in a sample, the new variance
a will be unchanged
b will be equal to the old variance plus the constant value.
c will be equal to the old variance plus the square of the constant value.
d none of these
4
In how many ways can you select from 7 people, 3 to serve on a committee?
a. 35
b. 24
c. 56
d. 5040
e. none of these
5
In testing a new drug, researchers found that 6% of all patients using it will have
a mild side effect. A random sample of 11 patients using the drug is selected. Find the
probability that none will have this mild side effect.
a 0.3063
b 0.9400
c 0.4937
d 0.0609
e 0.5063
6 Suppose you have a binomial distribution with n = 20 and p = 0.4. Find P(8 ≤ X ≤ 12).
a 0.3834
b 0.5955
c 0.9790
d 0.5631
e 0.9400
7 Which of the following is true?
a Binomial has a set number of trials
b Binomial has only success or fail
c Geometric has only success or fail
d Geometric does not have a set number of trials (we are looking for first success)
e All of the above are true
8 Joe has an 50% probability of passing his statistics quiz 4 each time he takes it. What
is the probability he will take no more than 5 tries to pass it?
a. 0.9844
b. 0.0156
c. 0.9688
d. 0.0034
e. none of these
9 Test is in CASA and need test reservation
a true
b false
10 Cannot be late for test
a true
b false