Density Curve
Density curve: Recall to determine the shape of a distribution, we drew a
fitted curve. This curve is known as the density curve
Density curve is ALWAYS on or above the horizontal axis
The total area under a density curve is ALWAYS equal

Binomial Distribution Formula
To find the probability of obtaining j success in n trials, we apply the
following formula:
j
n-j
P(X = j) = [(n! / (j!(n-j)!) (p (1-p) )]
Where (n! / j!(n-j)!) is known as the binomial coefficient and p
is the probability

Z-Score
Z-score: tells us how many standard deviations above or below the mean is
a particular value
Z = ( x - ) / )
Unit of measurement does not affect the z-score value
Z-scores have a mean of 0 and a standard deviation of 1
Most z-scores are betwe

Example
Due to decreasing revenue and economic conditions, many states have tried to
legalize gambling or, in states where gambling is already legal, expand casino
operations. To measure public opinion in Kansas, a random sample of residents
was selected

Empirical Rule for Sample
For any sample of observations with a symmetric and unimodal distribution, we expect to
find:
68% of all values fall within the interval (x-bar - s, x-bar + s)
95% of all values fall within the interval (x-bar - 2s, x-bar + 2s

Example for you to try
When we roll a die, there are six possible outcomes:
1, 2, 3, 4, 5, 6
Event A: The di come up even
2, 4, 6
Event B: The di comes up odd
1, 3, 5
Event C: The di comes up 6
6
List the outcomes in
(a) Not A (Ac)
1, 3, 5
(b)

Example of Linear Transformation
A city has recorded its daily maximum temperature during a month
Temperature (F)
Temperature (C)
Frequency
80.6
27
1
82.4
28
2
84.2
29
6
86
30
7
87.8
31
8
89.6
32
3
91.4
33
3
93.2
34
1
1 degree Celsius
1 x (9/5) + 32 =

Long Run Probability
In the short run, the probability fluctuates, but in the long run, it equals out
to 50%
Example
A test is consisted of two true or false problems, Suppose the answers to
both questions are true. Event A: Student gets exactly one out

Sections 1.1 and 1.2
Variability: Expected and signal
Experimental versus observational study
Identify an experimental/observational study
Observational (you don't set guidelines)
Experimental (you set guidelines)
Identify the experimental units, tr

Range
Range: The largest observation minus the smallest observation
Maximum - minimum = Range
If max was 80 and min was 60, then range would be 20
Range shows how spread out the data are
Range is not a resistant
Your extreme values (max or min) dire

Boxplot
A graph that displays the distribution of a data set using the five-number
summary
From a boxplot, we can easily see the outlier(s)
There is no limit to the amount of outliers we can have
Go from largest in or smallest in to figure out where t

Mode
Mode: The value that occurred the most often in the data set. To be considered
the mode, a value must appear at least 2 times (frequency >1)
Mode can be easily obtained once we construct the frequency distribution
It is possible for a data set to

Why Study Statistics?
Statistics provides us with an organized method for analyzing data in the
face of variability and uncertainty
Noise v. Signal
Variability in data is inevitable, but it is important to tell the difference
between noise and signal
N

Finding Probabilities Using a Normal Curve
Transform a given normal distribution with mean and standard deviation
into a standard normal distribution
To complete the transformation, for each value x, we perform the following
operations:
Z = [(x - ) /

More Example
The Hard Rock Caf in Dallas wants to monitor customer orders. Exactly half
of the customers ask for some kind of soda (S), while the other half order a
specialized tea (T). Suppose three customers are selected at random. Let
the random varia

Assignment 23
10.2.7
Condition
Seizure-free
Not Seizure-free
Total
Valproate
6
(6.49)
14
(13.51)
20
Phenytoin
6
(5.51)
11
(11.49)
17
Total
12
25
37
Assumptions
1. All expected frequencies are 1 or greater
2. At most 20% of the expected frequencies are les

PART A
Make sure to include all of the following elements in hypothesis tests:
State the null and alternative hypotheses in words and using proper symbols
If hypothesis is directional, check directionality first.
Determine and compute the appropriate t

Assignment 22
9.4.5
Let WF denote white feathers, DF denote dark feathers, SC denote small comb, and LC denote large comb.
Assumptions:
1. All expected frequencies are 1 or greater
2. At most 20% of the expected frequencies are less than 5.
3. Simple Rand

Assignment 21
9.1.3
a. Let success be the number of mutants
~
Y
Probability
p
0
1
2
3
4
5
y2 02 2
n4 54 9
3/9
4/9
5/9
6/9
7/9
Pr(Y j ) n C j p j 1 p n j
Pr(Y 0) 5 C0 0.37 0 1 0.37 50 0.10
0.29
0.54
0.20
0.06
0.01
~
Sampling distribution for P when p = 0.3

Assignment 17
7.2.2 a.
ts
ts
y1 y 2
2
2
s1 s 2
n1 n2
y1 y 2
where SE y1 y2
SE y1 y2
2
2
s1 s 2
n1 n2
100.2 106.8
5.7
t s 1.16
since t 0.1 1.372 and t 0.2 0.879, t he 0.20 p - value 0.40
b.
y y2
ts 1
SE y1 y2
ts
49.8 44.3
1.9
t s 2.89
since t 0.01

Assignment 18
7.3.5 If the null hypothesis is not rejected that the two samples are different, then the Type of Error would be
type II
7.3.8
a. HO: The new technique and the old technique have the same mean aging time.
HA: The new technique has less mean

Homework #19
7.6.3
Let 1 denote male and 2 denote female
Assumptions
1. Simple random samples
2. Normal Populations or large samples
3. Independent Samples
STEP 1: For a 95% confidence level use Table 4 to find t 0.025 with
df = df
SE SE .
SE SE
22
2
4

Assignment 20
8.2.2
a. The standard error for the mean difference would be
s
59.3
SEd d
19.77 lbs
n
9
b.
Assumptions
1. Simple random paired samples
2. Normal differences or large sample
STEP 1
HO: 1 2 OR 1 2 0 OR d 0
The mean weight gains on the two di

Assignment 16
6.6.3 SE(Y1Y2 ) 4.3
6.6.4 SE(Y1Y2 ) 3.04
6.6.5 SE(Y1 Y2 ) 5.90
6.6.9 SE(Y1 Y2 ) 10.2
6.7.2 SE(Y Y
1
2)
9.192 Dark is denoted as Sample 1 and Photoperiod is denoted as Sample 2.
a. For a 95% Confidence Interval
92 115 2.4479.192
43.5,0.5 or

*Note: All of these will have a "Areas under the standard normal curve"
table given to you*
Reading a Standard Normal Table
Standard normal table provides the area to the left of z-score under a
standard normal curve
Example, area under standard normal

Z-Score
Z-score: tells us how many standard deviations above or below the
mean is a particular value
Z = ( x - ) / )
Unit of measurement does not affect the z-score value
Z-scores have a mean of 0 and a standard deviation of 1
Most z-scores are betwe

*Note: All of these will have a "Areas under the standard normal curve" table
given to you*
Reading a Standard Normal Table
Standard normal table provides the area to the left of z-score under a
standard normal curve
Example, area under standard normal

STP 231 - Lecture 2.1, 2.2, and 2.3 - Different Ways to Organize Data
Variable
Variable: A characteristic of a person or a thing that can be assigned a
number or a category
Ex: Height, gender, and number of Facebook friends
A variable is NOT a specific