Download Top Ten #1

Top Ten #1
Descriptive Statistics
NOTE! This Power Point file is not
an introduction, but rather a checklist
of topics to review
Location: central tendency
• Population Mean =µ= Σx/N = (5+1+6)/3 =
12/3 = 4
• Algebra: Σx = N*µ = 3*4 =12
• Do NOT use if N is small and extreme
values
• Ex: Do NOT use if 3 houses sold this week,
and one was a mansion
Location
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Median = middle value
Ex: 5,1,6
Step 1: Sort data: 1,5,6
Step 2: Middle value = 5
OK even if extreme values
Home sales: 100K,200K,900K, so
mean =400K, but median = 200K
Location
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Mode: most frequent value
Ex: female, male, female
Mode = female
Ex: 1,1,2,3,5,8: mode = 1
Relationship
• Case 1: if symmetric (ex bell, normal), then
mean = median = mode
• Case 2: if positively skewed to right, then
mode<median<mean
• Case 3: if negatively skewed to left, then
mean<median<mode
Dispersion
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How much spread of data
How much uncertainty
Range = Max-Min > 0
But range affected by unusual values
Ex: Santa Monica = 105 degrees once a
century, but range would be 105-min
Standard Deviation
• Better than range because all data used
• Population SD = Square root of variance
=sigma =σ
• SD > 0
Empirical Rule
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Applies to mound or bell-shaped curves
Ex: normal distribution
68% of data within + one SD of mean
95% of data within + two SD of mean
99.7% of data within + three SD of mean
Sample Variance

( x  x)
n 1
2
Standard deviation = Square root
of variance
s

(x  x)
n 1
2
Sample Standard Deviation
X
6
6
7
8
13
Sum=40
Mean=40/5=8
xx
6-8=-2
6-8=-2
7-8=-1
8-8=0
13-8=5
Sum=0
( x  x )2
(-2)(-2)= 4
4
(-1)(-1)= 1
0
(5)(5)= 25
Sum = 34
Standard Deviation
Total variation = 34
• Sample variance = 34/4 = 8.5
• Sample standard deviation =
square root of 8.5 = 2.9
Graphical Tools
• Line chart: trend over time
• Scatter diagram: relationship between two
variables
• Bar Chart: frequency for each category
• Histogram: frequency for each class of
measured data (graph of frequency distr)
• Box Plot: graphical display based on
quartiles, which divide data into 4 parts
Top Ten #2
• Hypothesis Testing
Ho: Null Hypothesis
• Population mean=µ
• Population proportion=π
• Never include sample statistic in hypothesis
HA: Alternative Hypothesis
• ONE TAIL ALTERNATIVE
– Right tail: µ>number(smog ck)
π>fraction(%defectives)
Left tail: µ<number(weight in box of crackers)
π<fraction(unpopular President’s %
approval low)
Two-tail Alternative
• Population mean not equal to number (too
hot or too cold)
• Population proportion not equal to
fraction(% alcohol too weak or too strong)
Reject null hypothesis if
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Absolute value of test statistic > critical value
Reject Ho if |Z Value| > critical Z
Reject Ho if | t Value| > critical t
Reject Ho if p-value < significance level (note that
direction of inequality is reversed)
• Reject Ho if very large difference between sample
statistic and population parameter in Ho
Example: Smog Check
• Ho: µ = 80
• HA: µ > 80
• If test statistic =2.2 and critical value =
1.96, reject Ho, and conclude that the
population mean is likely > 80
• If test statistic = 1.6 and critical value =
1.96, do not reject Ho, and reserve
judgment about Ho
Type I vs Type II error
• Alpha=α = P(type I error) = Significance level =
probability you reject true null hypothesis
• Ex: Ho: Defendant innocent
• α = P(jury convicts innocent person)
• Beta= β = P(type II error) = probability you do not
reject a null hypothesis, given Ho false
• β =P(jury acquits guilty person)
Type I vs Type II Error
Reject Ho
Ho true
Ho false
Alpha =α =
P(type I error)
1–β
Do not reject Ho 1-α
Beta =β =
P(type II error)
Top Ten #3
• Confidence Intervals:
Mean and Proportion
Confidence Interval: Mean
•
Use normal distribution (Z table if):
population standard deviation (sigma)
known and either (1) or (2):
(1) Normal population
(2) Sample size > 30
Confidence Interval: Mean
• If normal table, then µ =(Σx/n)+ Z(σ/n1/2),
where n1/2 is the square root of n
Normal table
• Tail = .5(1 – confidence level)
• NOTE! Different statistics texts have
different normal tables
• This review uses the tail of the bell curve
• Ex: 95% confidence: tail = .5(1-.95)= .025
• Z.025 = 1.96
Example
• n=49, Σx=490, σ=2, 95% confidence
• µ = (490/49) + 1.96(2/7) = 10 + .56
• 9.44 < µ < 10.56
Conf. Interval: Mean
t distribution
• Use if normal population but population
standard deviation (σ) not known
• If you are given the sample standard
deviation (s), use t table, assuming normal
population
• If one population, n-1 degrees of freedom
t distribution
• µ = (Σx/n) + tn-1(s/n1/2)
Conf. Interval: Proportion
• Use if success or failure
(ex: defective or ok)
Normal approximation to binomial ok if
(n)(π) > 5 and (n)(1-π) > 5, where
n = sample size
π= population proportion
NOTE! NEVER use the t table if proportion!!
Confidence Interval: proportion
• Π= p + Z(p(1-p)/n)1/2
• Ex: 8 defectives out of 100, so p = .08 and
n = 100, 95% confidence
Π= .08 + 1.96(.08*.92/100)1/2
= .08 + .05
Interpretation
• If 95% confidence, then 95% of all
confidence intervals will include the true
population parameter
• NOTE! Never use the term “probability”
when estimating a parameter!! (ex: Do NOT
say ”Probability that population mean is
between 23 and 32 is .95” because
parameter is not a random variable)
Point vs Interval Estimate
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Point estimate: statistic (single number)
Ex: sample mean, sample proportion
Each sample gives different point estimate
Interval estimate: range of values
Ex: Population mean = sample mean + error
Parameter = statistic + error
Width of Interval
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Ex: sample mean =23, error = 3
Point estimate = 23
Interval estimate = 23 + 3, or (20,26)
Width of interval = 26-20 = 6
Wide interval: Point estimate unreliable
Wide interval if
• (1) small sample size(n)
• (2) large standard deviation(σ)
• (3) high confidence interval (ex: 99% confidence
interval wider than 95% confidence interval)
If you want narrow interval, you need a large sample
size or small standard deviation or low confidence
level.
Top Ten #4: Linear Regression
• Regression equation: y=bo+b1x
• y=dependent variable=predicted value
• x= independent variable
• bo=y-intercept =predicted value of y if x=0
• b1=slope=regression coefficient
=change in y per unit change in x
Slope vs correlation
• Positive slope (b1>0): positive correlation
between x and y (y incr if x incr)
• Negative slope (b1<0): negative correlation
(y decr if x incr)
• Zero slope (b1=0): no correlation(predicted
value for y is mean of y), no linear
relationship between x and y
Simple linear regression
• Simple: one independent variable, one
dependent variable
• Linear: graph of regression equation is
straight line
Coefficient of determination
• R2 = % of total variation in y that can be
explained by variation in x
• Measure of how close the linear regression
line fits the points in a scatter diagram
• R2 = 1: max possible value: perfect linear
relationship between y and x (straight line)
• R2 = 0: min value: no linear relationship
example
• Y = salary (female manager, in thousands of
dollars)
• X = number of children
• n = number of observations
Given data
x
y
2
48
1
52
4
33
Totals
x
y
2
48
1
52
4
33
Sum=7
Sum=133
n=3
7
x   2 .3
3
133
y
 44.3
3
Slope = -6.500
• Method of Least Squares formulas not on
301 exam
• B1 = -6.500 given
Interpret slope
If one female manager has 1 more
child than another, salary is $6500
lower
Intercept
bo= y – b1x
Intercept
bo=44.33-(-6.5)(2.33) = 59.5
Interpret intercept
If number of children is zero,
expected salary is $59,500
Regression Equation
• Y = 59.5 – 6.5X
Forecast salary if 3 children
59.5 –6.5(3) = 40
$40,000 = expected salary
y  average
y  actual
yˆ  forecast  bo  b1x
error  y  yˆ
S 

( y  yˆ )
n2
2
Standard error
(1)=x
(2)=y
48
(3) ŷ
(4)=
59.5-6.5x (2)-(3)
46.5
1.5
2
2.25
1
52
53
-1
1
4
33
33.5
-.5
.25
( y  yˆ ) 2
SSE=3.5
3.5
S 
 3.5  1.9
3 2
Interpret
Actual salary typically $1900 away
from expected salary
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Sources of Variation (V)
Total V = Explained V + Unexplained V
SS = Sum of Squares = V
Total SS = Regression SS + Error SS
SST = SSR + SSE
SSR = Explained V, SSE = Unexplained
Coefficient of Determination
• R2 = SSR
SST
• R2 = 197 = .98
200.5
• Interpret: 98% of total variation in salary
can be explained by variation in number of
children
0<
2
R
<1
• 0: No linear relationship since SSR=0
(explained variation =0)
• 1: Perfect relationship since SSR = SST
(unexplained variation = SSE = 0), but does
not prove cause and effect
R=Correlation Coefficient
• Case 1: slope < 0
• R<0
• R is negative square root of coefficient of
determination
R R
2
Our Example
• Slope = b1 = -6.5
• R2 = .98
• R = -.99
Case 2: Slope > 0
• R is positive square root of coefficient of
determination
• Ex: R2 = .49
• R = .70
• R has no interpretation
• R overstates relationship
Caution
• Nonlinear relationship (parabola, hyperbola,
etc) can NOT be measured by R2
• In fact, you could get R2=0 with a nonlinear
graph on a scatter diagram
R=correlation coefficient
• Case 1: If b1>0, R is the positive square
root of the coefficient of determination
• Ex#1: y = 4+3x, R2=.36: R = +.60
• Case 2: If b1<0, R is the negative square
root of the coefficient of determination
• Ex#2: y = 80-10x, R2=.49: R = -.70
• NOTE! Ex#2 has stronger relationship, as
measured by coefficient of determination
Extreme Values
• R=+1: perfect positive correlation
• R= -1: perfect negative correlation
• R=0: zero correlation
Top Ten #5
• Expected Value = E(x) = ΣxP(x)
= x1P(x1) + x2P(x2) +…
Expected value is a weighted average, also a
long-run average
E(x) Example
• Find the expected age at high school
graduation if 11 were 17 years old, 80 were
18, and 5 were 19
• Step 1: 11+80+5=96
Step 2
x
P(x)
xP(x)
17
11/96=.115
17(.115)=1.955
18
80/96=.833
18(.833)=14.994
19
5/96=.052
19(.052)=.988
E(x)=
17.937
Top Ten #6
• What distribution to use?
Use binomial distribution if:
• Random variable (x) is number of successes in n
trials
• Each trial is success or failure
• Independent trials
• Constant probability of success (π) on each trial
• Sampling with replacement (in practice, people
may use binomial w/o replacement, but theory is
with replacement)
Success vs failure
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Male vs female
Defective vs ok
Yes or no
Pass (8 or more right answers) vs fail (fewer
than 8)
• Buy drink (21 or over) vs can’t buy drink
Binomial is discrete
• Integer values
• 0,1,2,…n
• Binomial is often skewed, but may be
symmetric
Normal Distribution
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Continuous, bell-shaped, symmetric
Mean=median=mode
Measurement (dollars, inches, years)
Cumulative probability under normal curve : use Z
table if you know population mean and population
standard deviation
• Sample mean: use Z table if you know population
standard deviation and either normal population or
n > 30
t distribution
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Continuous, bell-shaped, symmetric
Applications similar to normal
More spread out than normal
Use t if normal population but population
standard deviation not known
• Degrees of freedom = df = n-1 if estimating
the mean of one population
• t approaches z as df increases
Top Ten #7
• P-value = probability of getting a sample
statistic as extreme (or more extreme) than
the sample statistic you got from your
sample
P-value example: 1 tail test
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Ho: µ = 40
HA: µ > 40
Sample mean = 43
P-value = P(sample mean > 43, given Ho
true)
• Reject Ho if p-value < α (significance level)
Two cases
• Suppose α = .05
• Case 1: p-value = .02, then reject Ho
(unlikely Ho is true; you believe population
mean > 40)
• Case 2: p-value = .08, then do not reject Ho
(Ho may be true; you have reason to believe
that population mean may be 40)
P-value example: 2 tail test
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Ho: µ = 70
HA: µ not equal to 70
Sample mean = 72
If 2-tails, then P-value =
2*P(sample mean > 72)=2(.04)=.08
If α = .05, p-value > α, so do not reject Ho
Top Ten #8
• Variation creates uncertainty
No variation
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Certainty, exact prediction
Standard deviation = 0
Variance = 0
All data exactly same
Example: all workers in minimum wage job
High variation
• Uncertainty, unpredictable
• High standard deviation
• Ex#1: Workers in downtown L.A. have
variation between CEOs and garment
workers
• Ex#2: New York temperatures in spring
range from below freezing to very hot
Comparing standard deviations
• Temperature Example
• Beach city: small standard deviation (single
temperature reading close to mean)
• High Desert city: High standard deviation
(hot days, cool nights in spring)
Standard error of the mean
• Standard deviation of sample mean =
standard deviation/square root of n
Ex: standard deviation = 10, n =4, so standard
error of the mean = 10/2= 5
Note that 5<10, so standard error < standard
deviation
As n increases, standard error decreases
Sampling Distribution
• Expected value of sample mean =
population mean, but an individual sample
mean could be smaller or larger than the
population mean
• Population mean is a constant parameter,
but sample mean is a random variable
• Sampling distribution is distribution of
sample means
Example
• Mean age of all students in the building is
population mean
• Each classroom has a sample mean
• Distribution of sample means from all
classrooms is sampling distribution
Central Limit Theorem
• If population standard deviation is known,
sampling distribution of sample means is
normal if n > 30
• CLT applies even if original population is
skewed
Top Ten #9
• Population vs sample
Population
• Collection of all items(all light bulbs made
at factory)
• Parameter: measure of population
(1)population mean(average number of
hours in life of all bulbs)
(2)population proportion(% of all bulbs that
are defective)
Sample
• Part of population(bulbs tested by inspector)
• Statistic: measure of sample = estimate of
parameter
(1) sample mean(average number of hours
in life of bulbs tested by inspector)
(2) sample proportion(% of bulbs in sample
that are defective)
Top Ten #10
• Qualitative vs quantitative
Qualitative
• Categorical data
success vs failure
ethnicity
marital status
color
zip code
4 star hotel in tour guide
Qualitative
• If you need an “average”, do not calculate
the mean
• However, you can compute the mode
(“average” person is married, buys a blue
car made in America)
Quantitative
• 2 cases
• Case 1: discrete
• Case 2: continuous
Discrete
(1) integer values (0,1,2,…)
(2) example: binomial
(3) finite number of possible values
(4) counting
(5) number of brothers
(6) number of cars arriving at gas station
Continuous
• Real numbers, such as decimal values
($22.22)
• Examples: Z, t
• Infinite number of possible values
• Measurement
• Miles per gallon, distance, duration of time
Graphical tools
• Pie chart or bar chart: qualitative
• Joint frequency table: qualitative (relate
marital status vs zip code)
• Scatter diagram: quantitative (distance from
CSUN vs duration of time to reach CSUN)
Hypothesis testing
Confidence intervals
• Quantitative: Mean
• Qualitative: Proportion