Midterm 1
MATH 161A Fall 2014 Name V
Professor Gottlieb LD.
Please answer all of the following questions. Show all your work. Partial credit can only be given
if your work can be followed. Please simply your answers. Point values are given below.
Total p
Mathematics 161A-Applied Statistics I
Lecture 1
Andrea Gottlieb
Greensheet
Instructor: Andrea Gottlieb
Oce Hours: Monday: 4:45pm-5:45pm, 7:15pm-7:45pm
Wednesday 10:45am-11:45am, 7:15pm-7:45pm
Email: [email protected]
Oce: MacQuarrie Hall 415
Lectur
Mathematics 161A-Applied Statistics I
Lecture 20
Andrea Gottlieb
1
Numerical vs Graphical Methods
So far we have covered three types of graphical displays of data
Histograms
Bar Charts
Boxplots
These graphical displays are a great way to obtain a quick
Midterm 3
, LVN; R if
MATH 161A Fall 2014 Name a r
Professor Gottlieb LD.
Section
Please answer all of the following questions. Show all your work. Partial credit can only be given if
your work can be followed. Please fully simply your answers. Total poi
Midterm 2
MATH 161A Fall 2014 Name @
Professor Gottlieb 1.1).
Please answer all of the following questions. Show all your work. Partial credit can only be given
if your work can be followed. Please simply your answers. Point values are given below.
Total
x) = 7'5 for '-5 5x S 5 and = 0 otherwise -
a. P(X<0)= fsfadx=.5.
2.
b. P(2.5<X<2.5)= [S-Idx:.5.
2.5lo
' 3
c. P(-2SXS3)= [210x=.5.
k+4
k _.
d; P(k<X<k+4)=J:+4T6dx=-l%x] 75[(k+4)k]=.4. . _ I 3 2
" P) 0): f-09375(4x2)dx=.09375[4xj_] =.5.
This matches the sy
Mathematics 161A-Applied Statistics I
Lecture 16
Andrea Gottlieb
1
Joint Probability Distributions
So far, we have studied probability models for a single random
variable. We have computed probabilities, expected values,
medians and variances for a single
46.
a. b(3;8,.35) = [gsscssf = .279.
b. b(5;8,.6) = [:](.6)5(.4)3= .279.
c. P(3 as 5) = b(3;7,.6) + b(4;7,.6) + b(5;7,.6) = .745.
d. PO 5):) = 1 -P(X= 0)=1[:](.1)(.9)9 = 1(.9)= .613. 49.
Rn
Let X be the number of seconds, so X ~ Bin(6, J 0)-
6
a. P(X= I):
Mathematics 161A-Applied Statistics I
Lecture 2
Andrea Gottlieb
1
Probability
Probability is used as a mathematical tool to understand or
describe random phenomena, chance variation and
uncertainty.
We will use probability as a tool to evaluate the reli
Mathematics 161A-Applied Statistics I
Lecture 19
Andrea Gottlieb
1
Statistics
So far, we have studied the topic of Probability, a branch of
mathematics concerned with the study of random phenomena.
Today we will begin to study Statistics, the science of c
Mathematics 161A-Applied Statistics I
Lecture 21
Andrea Gottlieb
1
Random Samples and Statistics
Denition: The RV X1 , X2 , . . . , Xn are said to form a (simple)
random sample of size n if
1. The Xi s are independent RVs.
2. Every Xi has the same probabi
Mathematics 161A-Applied Statistics I
Lecture 22
Andrea Gottlieb
1
Sampling Distributions
Numerical quantities calculated from the sample are called
statistics.
Statistics vary from sample to sample and hence are random
variables.
The probability distr
Mathematics 161A-Applied Statistics I
Lecture 23
Andrea Gottlieb
1
So Far.
Point Estimate: A single best guess for an unknown
population parameter
Condence Intervals: A range of plausible values for the
unknown population parameter
2
Hypothesis Testing
Mathematics 161A-Applied Statistics I
Lecture 24
Andrea Gottlieb
1
Steps for Conducting a Hypothesis Test
1. Identify the parameter of interest in the problem (typically
or p)
2. State the Null Hypothesis
3. State the Alternative Hypothesis (Decide if it
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a. I(.\" I, 3' l)p(l.i).20. I
b. P('.\'S i and Y5 I) (0.0) +110. HUvo) 0' 5
c. At least one hose is in use at both islands. 1"(X :6 0 and Y at 0) =P(1J)+P(L2) +P(2.1)
d. By summing row probabilities, pix) 1 .l6, .34, .50 forx = 0, l, 2, By sum
Mathematics 161A-Applied Statistics I
Lecture 3
Andrea Gottlieb
1
Probability Vocabulary:
Experiment: an action or process whose outcome is subject to
uncertainty.
Sample Space: denoted S is the set of all possible outcomes
of the experiment.
Event: a
Mathematics 161A-Applied Statistics I
Lecture 6
Andrea Gottlieb
1
Conditional Probability
A conditional probability of an event is the probability that the
event occurs given that some other event has occurred.
Conditional probabilities sometimes seem to
Mathematics 161A-Applied Statistics I
Lecture 4
Andrea Gottlieb
1
Sample Space with Equally Likely Simple Events
In many experiments consisting of N outcomes, it is reasonable to
assign equal probabilities to all N simple events. In this case
1
p = N.
Exa
Mathematics 161A-Applied Statistics I
Lecture 7
Andrea Gottlieb
1
Bayes Rule
Let B1 , . . . , Bk form a partition of the sample space S. Then for
any event A,
P(Bj |A) =
P(A|Bj )P(Bj )
k
i=1 P(A|Bi )P(Bi )
Note in the special case of the partition consist
Mathematics 161A-Applied Statistics I
Lecture 8
Andrea Gottlieb
1
Random Variables
In many random experiments we are not interested in the specic
outcome of an experiment, but in some characteristic of that
outcome. If you play the lottery, for example, y
Mathematics 161A-Applied Statistics I
Lecture 9
Andrea Gottlieb
1
Random Variables
For a given sample space of some experiment, a random variable
(RV) is any rule that associates a number with each outcome in S.
In mathematical language, a random variable
Mathematics 161A-Applied Statistics I
Lecture 10
Andrea Gottlieb
1
Binomial Distribution
Many discrete random variables fall into similar categories. For
example, from a mathematical point of view, the number of heads
in four tosses of a fair coin and the
Mathematics 161A-Applied Statistics I
Lecture 12
Andrea Gottlieb
1
Example 2: Hurricanes
Researchers from University College, London, UK, have studied the
hurricane activity in the Atlantic ocean. The annual hurricane
season lasts from June 1st through No
Mathematics 161A-Applied Statistics I
Lecture 11
Andrea Gottlieb
1
An Introduction to the Hypergeometric Random Variable
Situation: Suppose you have a population of size N, in which each
individual can be classied as either a success (S) or a failure (F).
Mathematics 161A-Applied Statistics I
Lecture 13
Andrea Gottlieb
1
Example:
Let X be a continuous random variable with PDF
f (x) =
2
9x
0
0x 3
otherwise
(d) Find the CDF of X .
(e) Draw a labeled sketch of the CDF of X .
(f) Use the CDF to compute P(X > 2
Mathematics 161A-Applied Statistics I
Lecture 14
Andrea Gottlieb
1
The Exponential Distribution
Remember: The Geometric distribution was used to model the
number of independent trials until (and including) the rst success.
The Exponential distribution is
Mathematics 161A-Applied Statistics I
Lecture 15
Andrea Gottlieb
1
The Normal Distribution
The Normal Distribution is by far the most important distribution
that we will study in this course. It is the distribution most often
applied to data by scientists
Mathematics 161A-Applied Statistics I
Lecture 17
Andrea Gottlieb
1
Conditional Distributions
When problems contain more than one random variable, and the
random variables are not independent, conditional probabilities can
become of special interest.
Suppo