3 Inferential Statistics Osychology.pptx

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 2-2
Statistics:
Descriptive and Inferential

Descriptive and Inferential

Inferential Statistics
 In inferential statistics and hypothesis testing, our goal
is to find systematic reasons for differences and rule out
random chance as the cause.
 Consider crop yield vs fertilizer use.
 For a given quantity of fertilizer (10 grm) crop yield
varies (20, 30, 40 Kg) though it should be one
number.
 The variation is due to many other factors that are
random (water, temperature etc.).
 We want to find the true (systematic) relationship
between fertilizer and yield and eliminates random
causes.

Relationships

Measuring Relationship
Between 2 Variables
Coefficient of Correlation
 Measures the relative strength of the linear
relationship between two variables
 Sample coefficient of correlation:
where
Y
X S
S
Y)
,
(X
cov
r 
1
n
)
X
(X
S
n
1
i
2
i
X





1
n
)
Y
)(Y
X
(X
Y)
,
(X
cov
n
1
i
i
i






1
n
)
Y
(Y
S
n
1
i
2
i
Y






R between depression and anxiety

Estimating r
Y
X S
S
Y)
,
(X
cov
r 

Features of
Correlation Coefficient, r
 Unit free
 Ranges between –1 and 1
 The closer to –1, the stronger the negative linear
relationship
 The closer to 1, the stronger the positive linear
relationship
 The closer to 0, the weaker the linear relationship
 R gives:
 form (linear), direction (+ or -) and magnitude (-1 to +1)

Scatter Plots of Data with Various
Correlation Coefficients
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
r = +.3
r = +1
Y
X
r = 0

Correlation between family size and stress
3 Correlation Family size & stress levels.xlsx
Family Size
Sress Level (0-
100)
Family Size 1
Stress Level
(0-100) 0.68 1

Pearson vs Spearman Correlation
 Pearson Correlation Coefficient (r):
 Pearson correlation coefficient measures the linear relationship between
two continuous variables.
 It assesses the degree to which the relationship between the variables can
be described by a straight line.
 Pearson correlation assumes that the variables are normally distributed
and have a linear relationship.
 Suitable for continuous data.
 Spearman Rank Correlation Coefficient (ρ or rs):
 Spearman rank correlation coefficient measures the strength and direction
of association between two variables, regardless of the linearity of the
relationship.
 It is suitable for both continuous and ordinal variables.
 It is robust to outliers and non-normal distributions.

Hypothesis Testing and
Relationship Testing
 Hypothesis testing
 T-Test (2 variables)
 ANOVA (several variables)
 Relationship testing
 Regression (dependent vs in dependent variables)

Simple

Multiple

Dummy

Binary dependent …

Intuition on Test of Hypothesis

 Purpose of inferential statistical analysis:
 To use sample data and make inferences about
what's truly (saying with confidence/ objectively)
happening in the population.
 We will attempt to understand intuitively the
theoretical basis on how we achieve above
purpose.

Types of Inferences
 1. Estimation and inferring of population parameters through a
sample.
 Estimate the average income of a Sri Lankan houshehold.
 2. Testing and inferring whether parameters of populations differ,
using samples.
 Test whether average household income between households
in Jaffna and Colombo districts are different.
 3. Regression inferring which independent variables significantly
explain a dependent variable and predicting the dependent
variable using independent variable/s, using a sample.
 What are the factors that explain household income?

Income f Education, Assets owned, political affiliations … Xn

ESTIMATION

Inferring on Mean of population from a sample

 Any phenomena has variation (thus referred to as
variable).
 Ex: household income of Sri Lankan population, academic
performance among CIRP students, stress levels among
CIRP students.
 Variation is that each individual score (observation)
will differ from its mean. Mean expected true value.
 Consider the income of households in Colombo and
observe above (variation, mean and difference of
observations to mean).
 6 Basis of statistical testing.xlsx

 Generate 1 random sample of the sample size 10 from a
population of 1 … 1868 Household Incomes in Colombo.
This data is from HIES (2019) Sri Lanka.
 Is your sample mean equal to the population mean?
 Ask your friends whether they got the same sample mean?
 Why is it that the sample mean not equal to population
mean?
 Just like an individual score will differ from its mean, an
individual sample mean will differ from the true population
mean.
 This deviation of sample mean from population mean is
called sampling error.

Sampling Distribution of Sample Means
 Take 10 more samples of 10 sample size on the population
of Colombo household income and estimate the sample
means for all 10 samples.
 Are the sample means the same? Why are they not the
same?
 Now estimate the mean of sample means (Distribution of
sample means) and check whether the mean of sample
means is equal to the population mean.
 It is getting closer to being equal to the population mean.
This is the basis of CENTRAL LIMIT THEOREM
 Note sampling distribution can be estimated for any
statistic as Mean, Correlation coefficient, T-stat etc.

CENTRAL LIMIT THEOREM
 The central limit theorem states:
 For samples of a single size n, drawn from a
population with a given mean μ and variance σ2, the
sampling distribution of sample means will have a
mean 𝜇𝑋 = μ and variance
̅ 𝜎𝑋
2
= σ2
/n.
 This distribution will approach normality as n
increases.
 Above is true even if the population distribution is not
normally distributed.

Implications of CLT
 Confidence Interval Estimation: The CLT is used to construct
confidence intervals for population parameters, such as the
population mean or proportion. Confidence intervals provide a
range of plausible values for the population parameter and are
widely used in inferential statistics to quantify the uncertainty
associated with sample estimates.
 Hypothesis Testing: The CLT is fundamental for hypothesis
testing, especially when dealing with large sample sizes. It justifies
the use of parametric tests, such as the t-test and z-test, even
when the population distribution is not normal, as long as the
sample size is sufficiently large. Hypothesis tests rely on the
assumption of normality or the CLT to make inferences about
population parameters.

Confidence Interval on
Sample Estimates
 6 Basis of statistical testing.xlsx CLT Simple spread
sheet
 Confidence Level: The confidence level is the
probability that the true population parameter falls
within the calculated confidence interval.
 i.e. 95% confidence level means that if one takes
many samples 95% of the standard errors of the
samples will contain the population mean. (pg 92
Gujarati)
 It is often expressed as a percentage (e.g., 95%
confidence level). Commonly used confidence levels
include 90%, 95%, and 99%.

Estimating a mean of a
population and reporting
 1. Take a sample (n > 30)
 2. Estimate sample mean and standard
error
 3. Report sample mean with standard
error
 If Mean +/- 2SE means that it is 95%
assured that the population mean is
between +/-2SE

TESTING

Normal Distribution

Converting ND to SND (Z)

Conversion to Z Value (Standard Normal Distribution)
 Any normal
distribution can be
converted to
Standard Normal
Distribution (Z
distribution).
 So the sampling
distribution can be
converted to a Z
distribution.

(Z Distribution) Has Known
Characteristics
 Mean (μ): The mean of
the Z-distribution is
always 0. This means
that the central value of
the distribution is located
at 0.
 Standard Deviation (σ):
The standard deviation of
the Z-distribution is
always 1.
 68% (probability) of the
observations will be
between +&- 1 SD etc.

Probability and Z values
 6 Basis of statistical testing.xlsx

Using Z distribution to
Test Hypothesis
 If a Z value is > 1.96
then probability of
that value coming
from that distribution
is less than 0.05.
 So if an estimated Z
value is more than
1.96 (say 2) then it is
not from that
distribution.

Z Value Table
 Using the z-table to find the area in the body to the left of z = 1.62

Example of Using the Z Distribution to
Test a Hypothesis
 Consider a psychologist, based on research knows that, the
average stress level of a person on a working day is 8.00
stress units. It is also known that the stress level varies by a
standard deviation σ = 0.50. That is, the known population
mean is μ = 8.00 and the known population standard
deviation is σ = 0.50.
 The psychologist wants to know (test) whether a new patient
is stressed. Hence he/she observes the patient for 25 days
(n=25) and takes measures of stress and found the average
stress to be = 7.75.
X
̅
 Test whether he is stressed?
 If stressed his stress level should fall within population stress levels
[with confidence level of 95%, i.e. estimated Z < 2 (from z table)]

Stating the Hypothesis
 Null hypothesis (H0) is typically the opposite of the researcher's hypothesis.
Null hypothesis is the idea that nothing is going on (no difference).
 H0 μ = 0
 Alternative hypothesis (HA ) is simply the reverse of the null hypothesis.
 There are three options, depending on where we expect the difference to lie
(direction).
 HA μ ≠ 0
 HA μ < 0
 HA μ > 0
 Can have numbers other than zero.
 Thus hypothesis on patients stress
 H0 μ = 8 (Stressed)
 HA μ < 8 (not stressed)
 Where μ = to patience average stress and 8 is population average stress.

Estimating the Z Score and
Testing Hypothesis
 Estimate the Z score and compare with table Z
value (for p=.05 Z is -1.96)
 Since Z estimated > Z table
 we reject the null hypothesis (patient stressed) and
accept the alternate hypothesis and conclude that
the patient is not stressed.
 Rule of thumb (first look at results)
 Estimated Z >2
 Estimated P < 0.05

T Distribution
 T distribution is used
when: Population
variance is not known
and sample size is
small.
 Different statistics
have different
distributions, which
are used for
hypothesis testing.

 Df= n-1 (n is sample size)

F Distribution (p=0.05)

Hypothesis Testing
Procedure
 Step 1: State the Hypotheses
 Step 2: Decide the Critical Values (99%, 95%,
the same p=0.01, p=0.05) and find the table
values.
 Step 3: Calculate the Test Statistic
 Step 4: Make the Decision: Compare the
estimated value with table value

Statistical Tests
7 Inferential Stat data.xlsx
 Test of means
 One sample (t test)

Two samples (t test)

Paired

Unpaired

Many samples (Anova F test)
 Test of difference of categorical measures
(frequency)
 Chi Square
 Regression: Test of association and quantification
of change

Single sample (called a 1-sample t-test)
 It’s known that patients with mean (µ)
depression scores > 30 needs clinical
attention. You have data of depression
scores of 4 check-ups (X) of a patient.
You want to infer whether the patient
needs clinical attention.
 Hypothesis: H0: X = 30 HA: X > 30
 Standard error:
 Standard deviation:

 Hypothesis:
 H0: X = 30 HA: X > 30
 Estimated t = 3.46
 Critical t = 2.35
 3.46 > 2.35
 Thus reject null
hypothesis
 Infer that the patient
needs treatment.
 Use STATA 7 Inferential Stat data.xlsx

Two sample paired t test
Depression scores/10
BeforeTrea
tment
After
Treatment Difference
Ranil 3 2 -1
Sajith 3 6 3
Anura 5 3 -2
Hirunika 8 4 -4
Namal 3 9 6
Patali 1 2 1
Diana 4 5 1
Mean 0.57
Std Dev 3.31
N 7
Sq Rt N 2.65
Std Error SD/SqRtN 1.25
t value 0.46
 Hypothesis:
 BT mean = AT mean
 Or BT mean-AT mean=0 or
 Mean of difference µD =0

Estimated t (0.46) < table t (2)

Two sample unpaired t test
 7 Inferential Stat data.xlsx

Test of difference of means of
many samples: ANOVA
 Scores on a job entrance test shown below, where applicants have different
qualifications. Want to infer whether different qualifications impact test result,

 Observe difference between groups (systematic variance) and also within groups
(unsystematic variance/random error). Our interest is to test the difference between
groups.

 Between group variance:
 Within group variance:
 Total Variance:

ANOVA Table: F interpretation
 If we are expecting a
large difference
between groups then
MSB > MSW
 i.e. a large F value
 Significance of F
value can be checked
over F Table.
 Hypothesis:
 Means of ND = RD= UR

 Hypothesis:
 H0: Means of ND = RD=
UR
 Consider table F = 2
 Estimated F (36) > table F (2)
 Reject H0
 Therefore infer there is
difference in performance
between degrees.
 We do not however know
the difference between
which degrees.
 7 Inferential Stat data.xlsx

F Table

Test of difference of categorical measures
(frequency measures): Chi Square (X2
) test
 Preference on pets given by a sample of people is given
below. The data is number of people preferring each
pet.
 We want to infer whether peoples preference differ.
 Expected values are what would be found if each
category had equal representation (no difference in
preference).

 Hypothesis: Pet
preference is equal
(not quantified)
 Estimated X2 is 6.49
 Table value is 5.99
 Est (6.49) > Tab
(5.99)
 Its inferred that there
is difference in pet
preference.

REGRESSION: TEST OF ASSOCIATION AND
QUANTIFICATION OF CHANGE

Regression: Test of association and
quantification of change
 Regression is estimating relationship (line) that “best fit” observations.
 How to estimate the line and how do we judge it’s the best fit?

 https://www.youtube.com/watch?v=PaFPbb66D
xQ

Estimating coefficients
 𝑐𝑜𝑣𝑋𝑌 is the covariance of X and Y we learned about with
correlations; and 𝑠𝑋
2
is the variance of X.
 Significance of b can be tested.
 Generally used estimation methods include Ordinary Least
Squares (OLS), Method of Moments (MoM), and Maximum
Likelihood Estimate (MLE).

Strength of the regression
 The "strength" of a regression model typically refers to how well the model fits the data or how
strong the relationship is between the independent variable(s) and the dependent variable.
There are several measures commonly used to assess the strength of a regression model:
 1. **Significance of Regression Coefficients**: The significance of individual regression
coefficients indicates whether each independent variable contributes significantly to the model's
prediction of the dependent variable. A higher significance level (typically p-value less than
0.05) suggests a stronger relationship.
 2. **Coefficient of Determination ((R^2))**: (R^2) measures the proportion of the variance in
the dependent variable that is explained by the independent variable(s) in the regression model.
It ranges from 0 to 1, where 1 indicates a perfect fit and 0 indicates no relationship. Higher
values of (R^2) indicate stronger relationships.
 3. **Adjusted (R^2)**: Adjusted (R^2) is a modified version of (R^2) that penalizes the
inclusion of additional independent variables that do not improve the model's fit. It adjusts for
the number of predictors in the model and is often preferred when comparing models with
different numbers of predictors.
 4. **F-statistic**: The F-statistic tests the overall significance of the regression model. A larger
F-statistic indicates a stronger relationship between the independent variable(s) and the
dependent variable.

Significance of Regression
Coefficients and signs
 regression coefficients is assessed through
hypothesis testing, typically using t-tests or
F-tests, and interpreting the associated p-
values. Significant coefficients indicate that
the corresponding independent variables
have a statistically significant impact on the
dependent variable.

Coefficient of Determination
R^2
 The coefficient of determination, often
denoted as �2R2, is a statistical measure
that represents the proportion of the
variance in the dependent variable that is
explained by the independent variables in a
regression model. It's a key metric used to
assess the goodness-of-fit of a regression
model.

Strength of the regression: F
Value
 SST (Total) the variance (deviation from
mean) we want to explain with the
regression line.
 SSR (Model) is deviation between models.
 SSE (Error) is the residual not explained by
the regression line. Which is to be
minimized. Then F will be large. The
significance of the F value can be tested.
Total Residual
Model

Estimating regressions and
test of hypothesis
 Simple regression 7 Inferential Stat data.xlsx
 Multiple regression
 Simple
 Multiple
 Categorical independent variables (dummy variable)
 Categorical dependent variable (Logistic/
Multinomial)
 …. ?

Parametric vs Nonparametric Tests
Parametric
 1. Assumptions:
 Parametric tests assume that the data are drawn from populations
with specific distributional characteristics, typically the normal
distribution.
 They also assume that the data have a specific level of measurement
(e.g., interval or ratio) and that the variances of the groups being
compared are equal.
 2. Examples: Parametric tests include:
 t-tests (e.g., independent samples t-test, paired samples t-test)
 Analysis of Variance (ANOVA)
 Pearson correlation
 Linear regression
 3. Advantages:
 Parametric tests are generally more powerful (i.e., have higher
statistical power) than nonparametric tests when the assumptions are
met.
 They often provide more precise estimates and smaller standard
errors.
 Parametric tests allow for more precise estimation of effect sizes
and confidence intervals.
 4. Limitations:
 Parametric tests are sensitive to violations of their assumptions. If
the data do not meet the assumptions (e.g., non-normality, unequal
variances), the results may be inaccurate.
 They may not be suitable for small sample sizes or data that do not
follow a normal distribution.
Nonparametric
 1. Assumptions:
 Nonparametric tests make fewer assumptions about the distribution
of the data. They are often referred to as distribution-free tests.
 They do not assume a specific probability distribution for the data
and are robust to violations of assumptions such as normality.
 2. Examples: Nonparametric tests include:
 Mann-Whitney U test (equivalent to the independent samples t-test)
 Wilcoxon signed-rank test (equivalent to the paired samples t-test)
 Kruskal-Wallis test (nonparametric alternative to ANOVA)
 Spearman rank correlation
 Dummy variables and logistic regression
 3. Advantages:
 Nonparametric tests are robust to violations of assumptions about
the distribution of the data.
 They can be used with ordinal or non-normally distributed data, as
well as with small sample sizes.
 Nonparametric tests are often simpler and more straightforward to
interpret.
 4. Limitations:
 Nonparametric tests are generally less powerful than parametric
tests when the assumptions of the latter are met.
 They may provide less precise estimates and wider confidence
intervals.
 Nonparametric tests may have lower efficiency (i.e., require larger
sample sizes) compared to parametric tests.

Degrees of freedom
 https://www.youtube.com/watch?v=92s7IVS6A3
4&t=2s
 https://www.youtube.com/watch?v=ke8nSbXUJj
Q
 So the degree of freedom is considered to give
more precision to estimate (standard deviation
etc.)

3 Inferential Statistics Osychology.pptx

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