Transcript Innovation in Planning Practice: A History of Metropolitan

URBP 204A
QUANTITATIVE METHODS I
Statistical Analysis Lecture III
Gregory Newmark
San Jose State University
(This lecture accords with Chapters 9,10, & 11 of Neil Salkind’s
Statistics for People who (Think They) Hate Statistics)
Statistical Significance Revisited
• Steps:
– State hypothesis
– Set significance level associated with null
hypothesis
– Select statistical test (we will learn these soon)
– Computation of obtained test statistic value
– Computation of critical test statistic value
– Comparison of obtained and critical values
• If obtained > critical reject the null hypothesis
• If obtained < critical stick with the null hypothesis
Three Statistical Tests
• t-Test for Independent Samples
– Tests between the means of two different groups
• t-Test for Dependent Samples
– Tests between the means of two related groups
• Analysis of Variance (ANOVA)
– Tests between means of more than two groups
t-Tests General Points
• Used for comparing sample means when
population’s standard deviation is unknown
(which is almost always)
• Accounts for the number of observations
• Distribution of t-statistic is identical to normal
distribution when sample sizes exceed 120
t-Tests General Points
t-Test of Independent Samples
• Compares observations of a single variable
between two groups that are independent
• Examples:
– “Are there differences in TV exposure between
teens in Oakland and San Francisco?”
– “We are going to take 100 people and give 50 of
them $2 and see which group is happier.”
– “In 2008, did the average visitor spend less time at
the art museum than at the planetarium?”
– “Do people in San Jose make different amounts of
monthly transit trips than folks in San Francisco?”
t-Test of Independent Samples
• Example:
– “Do people in San Jose make different amounts of
monthly transit trips than folks in San Francisco?”
• Steps:
– State hypotheses
• Null :
• Research :
H0 : µTrips San Jose = µTrips San Francisco
H1 : XbarTrips San Jose ≠ XbarTrips San Francisco
– Set significance level
• Level of risk of Type I Error = 5%
• Level of Significance (p) = 0.05
t-Test of Independent Samples
• Steps (Continued)
– Select statistical test
• t-Test of Independent Samples
– Computation of obtained test statistic value
• Insert obtained data into appropriate formula
• (SPSS can expedite this step for us)
t-Test of Independent Samples
• Formula
• Where
– Xbar is the mean
– n is the number of participants
– s is the standard deviation
– Subscripts distinguish between Groups 1 and 2
t-Test of Independent Samples
• Data
Trips San Jose
Trips San Francisco
7
8
6
4
2
5
8
5
5
6
3
5
10
3
5
4
8
4
6
2
3
8
10
5
2
4
9
4
4
8
2
5
5
7
12
5
8
6
3
9
3
5
1
1
15
5
3
7
2
7
8
4
1
9
4
7
2
7
7
6
San Jose
Mean = 5.43
n = 30
s = 3.42
San Francisco
Mean = 5.53
n = 30
s = 2.06
t-Test of Independent Samples
• Steps (Continued)
– Computation of obtained test statistic value
• tobtained = -0.14
• (don’t worry about the sign here)
– Computation of critical test statistic value
•
•
•
•
•
Value needed to reject null hypothesis
Look up p = 0.05 in t table
Consider degrees of freedom [df= n1 + n2 – 2]
Consider number of tails (is there directionality?)
tcritical = 2.001
t-Test of Independent Samples
• Steps (Continued)
– Comparison of obtained and critical values
• If obtained > critical reject the null hypothesis
• If obtained < critical stick with the null hypothesis
• tobtained = |-0.14| < tcritical = 2.001
– Therefore, we cannot reject the null hypothesis
and we thus conclude that there are no
differences in the mean transit trips per month
between people in San Jose and San Francisco
t-Test of Dependent Samples
• Compares observations of a single variable
between one group at two time periods
• Examples:
– “Does watching this movie make audiences feel
happier?”
– “Does a certain curriculum initiative improve
student test results?”
– “Do people make more transit trips with the
extension of a BART line to their neighborhood?”
– “Does sensitivity training make people more
sensitive?”
t-Test of Dependent Samples
• Example:
– “Does sensitivity training make people more
sensitive?”
• Steps:
– State hypotheses
• Null :
• Research :
H0 : µbefore training = µafter training
H1 : Xbarbefore training < Xbarafter training
– Set significance level
• Level of risk of Type I Error = 5%
• Level of Significance (p) = 0.05
t-Test of Dependent Samples
• Steps:
– Select statistical test
• t-Test of Dependent Samples
– Computation of obtained test statistic value
• Insert obtained data into appropriate formula
• (SPSS can expedite this step for us)
t-Test of Dependent Samples
• Formula
t-Test of Dependent Samples
Subject
Before
After
Difference
Difference2
1
3
7
4
16
2
5
8
3
9
3
4
6
2
4
4
6
7
1
1
5
5
8
3
9
6
5
9
4
16
7
4
6
2
4
8
5
6
1
1
9
3
7
4
16
10
6
8
2
4
11
7
8
1
1
12
8
7
-1
1
Sum
61
87
26
82
t-Test of Dependent Samples
• Steps (Continued)
– Computation of obtained test statistic value
• tobtained = 4.91
• (don’t worry about the sign here)
– Computation of critical test statistic value
•
•
•
•
•
Value needed to reject null hypothesis
Look up p = 0.05 in t table
Consider degrees of freedom [df = n -1 ]
Consider number of tails (is there directionality?)
tcritical = 1.80
t-Test of Dependent Samples
• Steps (Continued)
– Comparison of obtained and critical values
• If obtained > critical reject the null hypothesis
• If obtained < critical stick with the null hypothesis
• tobtained = |4.91| > tcritical = 1.80
– Therefore, we reject the null hypothesis and we
thus conclude that the sensitivity training works
Goodbye, t-Tests. Hello, ANOVA.
Simple ANOVA
• Compares observations of a single variable
between multiple groups
• Examples:
– “Are there differences between the reading skills
of high school, college, and graduate students?”
– “Does environmental knowledge vary between
people who commute by car, bus, and walking?”
– “Are there wealth differences between A’s, Giants,
Dodger, and Angels fans?”
– “Are there differences in the speech development
among three groups of preschoolers?”
Simple ANOVA
• Also called One-way ANOVA
• Compares means of more than two groups on
one factor or dimension with F statistic
• Calculated as a ratio of the amount of variability
between groups (due to the grouping factor) to
the amount of variability within groups (due to
chance)
–F=
Variability between different Groups
Variability within each Group
– As this ratio exceeds one it is more likely to be due
to something other than chance
• No directionality, therefore no issue of tails
Simple ANOVA
• Example:
– “Are there differences in the speech development
among three groups of preschoolers?”
• Steps:
– State hypotheses
• Null :
• Research :
H0 : µgroup 1 = µgroup 2 = µgroup 3
H1 : Xbargroup 1 ≠ Xbargroup 2 ≠ Xbargroup 3
– Set significance level
• Level of risk of Type I Error = 5%
• Level of Significance (p) = 0.05
Simple ANOVA
• Steps:
– Select statistical test
• Simple ANOVA
– Computation of obtained test statistic value
• Insert obtained data into appropriate formula
• (SPSS can expedite this step for us)
Simple ANOVA
• Formula
Sum Squares
between
MeanSum Squ
aresbetween
k 1
F

Sum Squares
MeanSum Squ
areswithin
within
N k
When : k  groups; N  cases
Simple ANOVA
Data
Group
1
Group
2
Group
3
3
2
1
4
3
1
5
2
1
5
3
1
5
2
1
4
1
1
4
1
1
3
1
1
4
1
1
5
1
1
n
10
10
10
Sum
42
17
10
Mean
4.2
1.7
1.0
• Fobtained = 65.31
• Degrees of Freedom
– Numerator = 2
– Denominator = 27
Simple ANOVA
• Steps (Continued)
– Computation of obtained test statistic value
• Fobtained = 65.31
– Computation of critical test statistic value
• Value needed to reject null hypothesis
• Look up p = 0.05 in F table
• Consider degrees of freedom for numerator and
denominator
• No need to worry about number of tails
• Fcritical = 3.36
Simple ANOVA
• Steps (Continued)
– Comparison of obtained and critical values
• If obtained > critical reject the null hypothesis
• If obtained < critical stick with the null hypothesis
• Fobtained = 65.31 > Fcritical = 3.36
– Therefore, we reject the null hypothesis and we
thus conclude that there are differences in the
speech abilities of the students in the preschools.