Principles of Statistics Assoc' Prof' Dr' Abdul Hamid b' Hj' Mar Iman Former Director, Centre for Re

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Principles of StatisticsAssoc. Prof. Dr. Abdul
Hamid b. Hj. Mar ImanFormer Director,Centre for
Real Estate StudiesFaculty of Geoinformation
Science and Engineering,Universiti Teknologi
Malaysia,Skudai, Johor.E-mail hamid_at_fksg.utm.my
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Hypothesis Testing

• Content
• Concepts of hypothesis testing
• Test of statistical significance
• Hypothesis testing one variable at a time

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Hypothesis

• Unproven proposition
• Supposition that tentatively explains certain
  facts or phenomena
• Assumption about nature of the world
• E.g. the mean price of a three-bedroom single
  storey houses in Skudai is RM 155,000.

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Hypothesis (contd.)

• An unproven proposition or supposition that
  tentatively explains certain facts or phenomena
• Null hypothesis
• Alternative hypothesis
• Null hypothesis is that there is no systematic
  relationship between independent variables (IVs)
  and dependent variables (DVs).
• Research hypothesis is that any relationship
  observed in the data is real.

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Null Hypothesis

• Statement about the status quo
• No difference
• Statistically expressed as
• Ho b0
• where b is any sample parameter used to
  explain the population.

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Alternative Hypothesis

• Statement that indicates the opposite of the null
  hypothesis
• There is difference
• Statistically expressed as
• H1 b ? 0
• H1 b lt 0
• H1 b gt 0

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Significance Level

• Critical probability in choosing between the Ho
  and H1.
• Simply means, the cut-off point (COP) at which a
  given value is probably true.
• Tells how likely a result is due to chance
• Most common level, used to mean something is
  good enough to be believed, is .95.
• It means, the finding has a 95 chance of being
  likely true.
• What is the COP at 95 chance?

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Significance Level (contd.)

• Denoted as ?
• Tells how much the probability mass is in the
  tails of a given distribution
• Probability or significance level selected is
  typically .05 or .01
• Too low to warrant support for the null
  hypothesis
• In other words, high chances to warrant support
  for alternative hypothesis
• Main purpose of statistical testing to reject
  null hypothesis

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Significance Level (contd.)
P-1.96 ? Z ? 1.96 1 - ? 0.95 PZ ? Zc
PZ ? -Zc ?/2
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• Let say we have the following relationship
  
• Y ß ei i1,, T and ei N(0,s2)
  ....(1)
• The least square estimator for ß is
• T
• b?Yi/T ..(2)
• i1
• with the following properties
• 1) Eb ß .(3a)
• 2) Var(b)E(b-ß)2 s2/T ...(3b)
• 3) bN(ß, s2/T) .(3c)

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• The standardized normal random variable for ß
  is
• b-ß
• Z -------- N(0,1) ..(4)
• ?(s2T)
• The critical value of Z, i.e. Zc, such that
  a0.05 of the
• probability mass is in the tails of distribution,
  is given as
• PZ? 1.96 PZ ?-1.960.025 (5a)
• and
• P-1.96 ? Z ?1.961-0.050.95 (5b)

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• Substituting SND for variable ß (Eqn. 4) into Eqn
  (5a),
• we get
• b-ß
• P-1.96 ? --------- ?1.960.95
  .....(6)
• ?(s2/T)
• Solving for ß, we get
• Pb-1.96s/?T ?ß ?b1.96s/?T0.95 (7)
• In general Pb-Zcs/?T ? ß ?bZcs/?T 1-?
  .. (8a)
• b-ß b -ß
• Also P------- ? -Zc P -------- ? Zc a/2
  (2-tail test) ...(8b)
• s/?T s/?T

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Example You suspect that the mean rental of 225
purpose-built office units in Johor is RM
3.00/sq.ft. If the std. dev. is RM 1.50/sq.ft.,
what is the 95 confidence interval of the mean?
The null hypothesis that the mean is equal to 3.0
Ho µ 3.0
The alternative hypothesis that the mean does not
equal to 3.0
H1 µ ? 3.0
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A Sampling Distribution
a.025
a.025
m3.0
-XL ?
XU ?

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Critical values of m
Critical value - upper limit
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Critical values of m
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Critical values of m
Critical value - lower limit
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Critical values of m
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Region of Rejection
LOWER LIMIT
UPPER LIMIT
m3.0

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Hypothesis Test m 3.0

2.804
3.78
3.196
m3.0
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Type I and Type II Errors
Accept null
Reject null
Null is true
Correct- no error
Type I error
Null is false
Type II error
Correct- no error
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Type I and Type II Errors in Hypothesis Testing
State of Null Hypothesis Decision in
the Population Accept Ho Reject Ho Ho is
true Correct--no error Type I error Ho is
false Type II error Correct--no error
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Example

• You estimate that the average price, µ, of
  single-
• and double-storey houses in Malaysias major
• industrialised towns to be RM 1,600/sq.m.
• Based on a sample of 101 houses, you found
• that the mean price, , is 1,579.44/sq.m. with
  a std
• dev. of RM 350.13/sq.m.
• Would you reject your initial estimate at 0.05
  significance level?
• What is the confidence interval of rental at 5
  s.l.?

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• Answer (a)
• Ho 1,600
• H1 ? 1,600
• 1,579.44 1,600
• Test statistic Z --------------------
• 350.13/?101
• -0.59
• PZ ? Zc PZ ? -Zc 0.05
• P0.59 ? Zc 0.05
• From Z-table, Zc 1.645
• Since Z lt Zc,do not reject Ho.
• ? Rental RM 1,600/sq.m.

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• Answer (b)
• 1,579.13-1.645(34.84)RM 1,521.82 (lower limit)
• 1,579.131.645(34.84)RM 1,636.44 (upper limit)

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NONPARAMETRIC STATISTICS
PARAMETRIC STATISTICS
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t-Distribution

• Symmetrical, bell-shaped distribution
• Mean of zero and a unit standard deviation
• Shape influenced by degrees of freedom

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Degrees of Freedom

• Abbreviated d.f.
• Number of observations
• Number of constraints

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Confidence Interval Estimate Using the
t-distribution
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Confidence Interval Estimate Using the
t-distribution
population mean sample mean critical
value of t at a specified confidence level
standard error of the mean sample standard
deviation sample size
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Confidence Interval Estimate Using the
t-distribution
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Hypothesis Test Using the t-Distribution
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Univariate Hypothesis Test Utilizing the
t-Distribution
Suppose that a production manager believes the
average number of defective assemblies each day
to be 20. The factory records the number of
defective assemblies for each of the 25 days it
was opened in a given month. The mean was
calculated to be 22, and the standard deviation,
,to be 5.
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Univariate Hypothesis Test Utilizing the
t-Distribution
The researcher desired a 95 percent confidence,
and the significance level becomes .05.The
researcher must then find the upper and lower
limits of the confidence interval to determine
the region of rejection. Thus, the value of t is
needed. For 24 degrees of freedom (n-1, 25-1),
the t-value is 2.064.
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Univariate Hypothesis Test t-Test
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Testing a Hypothesis about a Distribution

• Chi-Square test
• Test for significance in the analysis of
  frequency distributions
• Compare observed frequencies with expected
  frequencies
• Goodness of Fit

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Chi-Square Test
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Chi-Square Test
x² chi-square statistics Oi observed
frequency in the ith cell Ei expected frequency
on the ith cell
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Chi-Square Test Estimation for Expected Number
for Each Cell
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Chi-Square Test Estimation for Expected Number
for Each Cell
Ri total observed frequency in the ith row Cj
total observed frequency in the jth column n
sample size
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Univariate Hypothesis Test Chi-square Example
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Univariate Hypothesis Test Chi-square Example
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Hypothesis Test of a Proportion

• p is the population proportion
• p is the sample proportion
• p is estimated with p

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Hypothesis Test of a Proportion

p
5
.


H
0
¹
p
5
.


H
1
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Hypothesis Test of a Proportion Another Example
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Hypothesis Test of a Proportion Another Example
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Hypothesis Test of a Proportion Another Example
p
-
p

Z
S
p
-
15
.
20
.

Z
0115
.
05
.

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0115
.

348
.
4
Z
Indeed