Testing of hypothesis

Data Analysis Course
Testing of Hypothesis
Venkat Reddy

• Data analysis design document
• Introduction to statistical data analysis
• Descriptive statistics
• Data exploration, validation & sanitization
• Probability distributions examples and applications

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• Simple correlation and regression analysis
• Multiple liner regression analysis
• Logistic regression analysis

• Testing of hypothesis
• Clustering and decision trees
• Time series analysis and forecasting
• Credit Risk Model building-1 2
• Credit Risk Model building-2

Note
• This presentation is just class notes. The course notes for Data
Analysis Training is by written by me, as an aid for myself.
• The best way to treat this is as a high-level summary; the
actual session went more in depth and contained other

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information.
• Most of this material was written as informal notes, not
intended for publication
• Please send questions/comments/corrections to
venkat@trenwiseanalytics.com or 21.venkat@gmail.com
• Please check my website for latest version of this document
-Venkat Reddy 3

Contents
• What is the need of testing
• Recap of sampling distribution
• Hypothesis testing
• Five main steps in testing
• Assumptions
• Hypotheses

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• Test Statistic
• P-value (P)
• Conclusion:
• Testing example
• Types of errors
• Testing for Means
• Z test
• T test
• Testing for Proportions
• Test of independence 4

Inference
• A cake, (weighs 20 Kg) how do you decide about its taste? A piece of
cake or after eating completely
• A truck half filled with oranges rest with apples. How would you
verify that? By counting them?
• Apple & Samsung sell mobiles all over the world. If you want to find

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which one people prefer, Do you take the opinion poll from all the
users around the world?
• Product manager claims that girls are buying their product more
than boys? Is there any association between gender and buying or
not buying?

5

Inference
• A cake, (weighs 20 Kg) how do you decide about its taste? A piece of cake or
after eating completely
• You took a piece of cake, you are not completely satisfied with the taste –Would
you recommend the cake?
• You took a 10 gram piece of cake, you didn’t like the taste—What would you say?
• You took a 1 milligram piece of cake, you liked the taste—What would you say?

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• A truck half filled with oranges rest with apples (owner claims 10,000
oranges & 10,000 apples) . How would you verify that? By counting them?
• You randomly picked 200 fruits, 120 are apples and 80 are oranges. What would
you say?
• You randomly picked 200 fruits, 190 are apples and 10s are oranges. What would
you say?

How bad in the sample is bad enough to say the population is also
bad 6

Statistical Inference
• Inferences about a population are made on the basis of results
obtained from a sample drawn from that population
• Want to talk about the larger population from which the
subjects are drawn, not the particular subjects!

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• A hypothesis test is a process that uses
sample statistics to test a claim about the
value of a population parameter.
• A verbal statement, or claim, about a
population parameter is called a statistical
hypothesis.
• Hypothesis: Proportion of apples = 0.5
7

Applications of testing
• Law and Forensics
• Testing for discrimination (in admission, hiring, pay, promotion practices, etc.) –
test of proportions
• Paternity testing
• Testing whether evidence found on a suspect came from the crime scene (blood,
fiber, fingerprints, ...)

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• Indeed, testing whether the defendant is guilty or not
• Medicine and Health
• Testing if a new drug is effective or ineffective-test of association
• Testing if particulate matter in air pollution causes lung cancer
• Testing if a particular gene is responsible for hemophilia
• Industry/Business/Economics
• Testing if a production machine is ‘in control’ or not-test of means
• Testing if a silicon wafer is good for use
• Science and Engineering
• Testing which theory of gravitation is correct, based on dark matter 8
• Testing whether men and woman differ according to a psychological trait
• Testing if two species have a common ancestor

Hypothesis Testing – An example
CEO of SBI claims employees mean age in SBI bank is 35
(292,215 employees). How can we prove or disprove it?
1. Take a random sample (500) and find their age
2. If it is near 35 then we say there is no evidence to

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reject that hypothesis
3. What if sample average age is lower or higher than 35 ?
4. How far is really far? We want to quantify the severity
of deviation……
5. Lets find the probability of this occurrence
6. It if it is really low, then we say we reject null
hypothesis 9

Hypothesis testing process
Assume the
population
mean age is 35.
(Null Hypothesis)

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Population
100,000
The Sample
Is X  40   35? Mean Is 40
No, not likely!

REJECT

Null Hypothesis Sample 10
500

Reason for Rejecting H0
Sampling Distribution

... Therefore, we reject
the null hypothesis

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that  = 35.

... if in fact this were
the population mean.

 = 35 40
It is unlikely that we would get a 11
H0 sample mean of this value ...

Sampling distribution -Recap
• A sampling distribution is the probability distribution of a sample
statistic that is formed when samples of size n are repeatedly taken
from a population.
• If the sample statistic is the sample mean, then the distribution is
the sampling distribution of sample means.

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Sample

Sample

Sample Sample
Sample Sample

The sampling distribution consists of the values of the sample
12
means,

Central Limit theorem -Recap
If a sample n (30) is taken from a population with
any type distribution that has a mean =
and standard deviation =
the sample means will have a normal distribution
and standard deviation

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13

x

Five Step in Testing of Hypothesis
1. Make Assumptions and meet test requirements.

2. State the null hypothesis.

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3. Select the sampling distribution and establish the critical
region.

4. Compute the test statistic.

5. Make a decision and interpret results.
14

Step 1: Make Assumptions and Meet Test
Requirements
• Random sampling
• Hypothesis testing assumes samples were selected using random
sampling.
• In this case, the sample of 500 cases was randomly selected from
all major branches.

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• Level of Measurement is Interval-Ratio.
• Yes age is not a categorical variable
• Sampling Distribution is normal in shape.
• What is the sampling distribution of age?
• This is a “large” sample (n≥100).

15

Step 2 State the Null Hypothesis
• H0: μ = 35
• In other words, Ho: No difference between the sample mean
and the population parameter
• In other words, The sample mean of 40 is really the same as

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the population mean of 35 – the difference is not real but is
due to chance.
• In other words, The sample of 500 comes from a population
that has average age of 35
• In other words, The difference between 35 and 40 is trivial
and caused by random chance.

16

Step 2 (cont.) State the Alternate Hypothesis

• H1: μ≠35
• Or H1: There is a difference between the sample mean and
the population parameter
• Or The sample of 500 comes a population that does not have

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average age 35 In reality, it comes from a different population.
• Or The difference between 40 and 35 reflects an actual
difference
• Or the average age of the population is more than 35

17

Step 3 Select Sampling Distribution and
Establish the Critical Region
• What is the sampling distribution of population mean?
• What is alpha?
• Probability of rejecting H0 when it is true
• α is the indicator of “rare” events.
• Any difference with a probability less than α is rare and will cause us

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to reject the H0.
• We started with H0 as true, we still want to reject the null if the
statistic is beyond a certain value,
• We already know about some unlikely values of test statistic when
null hypothesis is true
• for example if the average age of the sample is 60, we definitely
want to reject null

Details later 18

Step 4: Use Formula to Compute the Test
Statistic - Z for large samples (≥ 100)
• We got the sample average as 40, the age according to null hypothesis
is 35, there is a difference of 5, is it due to chance?
• How bad is this difference of 5?


Z

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 N
When the Population σ is not known, use the following formula:

 40  35
Z Z
s n 1 7.86 500  1
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Step 5 Make a Decision and Interpret
Results
• The obtained Z score fell in the Critical Region, so we reject the H0.
• If the H0 were true, a sample outcome of 14 would be unlikely.
• Therefore, the H0 is false and must be rejected.

a

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H0:   35
H1:  > 35 P-Value

0
What does z of 14 mean? The probability of z being more than 14 is less than
0.000000001
• It is like getting more than 25 heads in a row when you toss a coin
• It is like drawing the same card more than 6 times from a shuffled deck
• If the average age of 40 is just by chance then compare that chance with above 20
examples

What is P-Value
• If the observed statistic happens to be just a chance, p-values tells
us what is the probability of that chance
• The P-value answer the question: What is the probability of the
observed test statistic or one more extreme when H0 is true?
• Given H0, probability of the current value or extreme than this

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• Given H0 is true, probability of obtaining a result as extreme or more
extreme than the actual sample
• The observed significance level, or p-value of a test of hypothesis is
the probability of obtaining the observed value of the sample
statistic, or one which is even more supportive of the alternative
hypothesis, under the assumption that the null hypothesis is true.
• Smallest α the observed sample would reject H0
21

P -value
• If the alternative hypothesis contains the greater-than symbol (>), the
hypothesis test is a right-tailed test.

H0: μ =k
Ha: μ > k

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P is the area to
the right of the
test statistic.

z
-3 -2 -1 0 1 2 3
Test 22
statistic

One tail & two tailed tests
• Two-tailed Test
• If the alternative hypothesis contains the not-equal-to symbol (),
the hypothesis test is a two-tailed test. In a two-tailed test, each
tail has an area of 0.5P.
• H0: μ = k
• Ha: μ  k

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• Left-tailed Test
• If the alternative hypothesis contains the less-than inequality
symbol (<), the hypothesis test is a left-tailed test.
• H0: μ  k
• Ha: μ < k
• Right-tailed Test
• If the alternative hypothesis contains the less-than inequality
symbol (<), the hypothesis test is a left-tailed test.
• H0: μ  k 23
• Ha: μ < k

Types of errors
• No matter which hypothesis represents the claim, always
begin the hypothesis test assuming that the null hypothesis is
true.
• At the end of the test, one of two decisions will be made:

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• reject the null hypothesis, or
• fail to reject the null hypothesis.
• A type I error occurs if the null hypothesis is rejected when it
is true.
• A type II error occurs if the null hypothesis is not rejected
when it is false.

24

Error Types
• Type I Error: Reject H0 when it is true
• Type II Error: Do not reject H0 when it is false

Test Result – Don’t Reject

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Reject H0
H0
Reality
H0 True Type I Error Correct

H0 False Correct Type II Error

25

Level of Significance a
• Defines Unlikely Values of Sample Statistic if Null Hypothesis Is
True
• Called Rejection Region of Sampling Distribution
• Designated a (alpha)

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• Typical values are 0.01, 0.05, 0.10
• Selected by the Researcher at the Start Provides the Critical
Value(s) of the Test
• P(Type I error)
• Think of analogy with SBI average age

26

Level of Significance, a and the Rejection
Region

H0:   35 a Critical
Value(s)
H1:  < 35
0

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Rejection Regions
a
H0:   35
H1:  > 35 0
a/2
H0:   35
H1:   35
0 27

Power of test 1-b
• P(Type II error) is b
• P(Type II error) = b depends on the true value of the
parameter (from the range of values in Ha ).
• The farther the true parameter value falls from the null value,

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the easier it is to reject null, and P(Type II error) goes down.
• Power of test = 1 - b = P(reject null, given it is false)
• In practice, you want a large enough n for your study so that
P(Type II error) is small for the size of effect you expect.

28

Which error is bad?
• False negative
• Miss what could be important
• Testing a metal whether it is gold or not
• Are these samples going to be looked at again?
• False positive

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• Waste resources following dead ends
• Test whether a drug is deadly or not

29

Confidence Intervals
• Hypothesis testing focuses on where the sample mean is
located
• Confidence intervals focus on plausible values for the
population mean
• General Formula (1-α)% CI for μ

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 Z1a / 2 Z1a / 2 
X  ,X  
 n n 

• Construct an interval around the point estimate
• Look to see if the population/null mean is inside 30

Significance Test for Mean

For large samples Z
 N

y  0
For small samples t  where se  s / n
se

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• Sampling distribution for small samples is t
• The curve of the t distribution varies with sample size (the smaller the size, the
flatter the curve)
• In using the t-table, we use “degrees of freedom” based on the sample size.
• For a one-sample test, df = n – 1.
• When looking at the table, find the t-value for the appropriate df = n-1. This
will be the cutoff point for your critical region. 31

Lab: Significance Test for Mean
• It is known that the mean cholesterol level for the nation is 190. We
test 100 only children and find that the sample average cholesterol level is
198 and suppose we know the population standard deviation  = 15. does
that signify that only children have an average higher cholesterol level than
the national average?
• Given this sample what are the reasonable values for population mean?

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• 50 smokers were questioned about the number of hours they sleep each
day. We want to test the hypothesis that the smokers need less sleep than
the general public which needs an average of 7.7 hours of sleep. If the
sample mean is 7.5 and the population standard deviation is 0.5, what can
you conclude?
• Given this sample what are the 95% confidence limits for population mean?

32

Test of Proportion
• Assumptions:
• Categorical variable
• Randomization
• Large sample (but two-sided ok for nearly all n)

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• Hypotheses:
• Null hypothesis: H0: p  p0
• Alternative hypothesis: Ha: p  p0 (2-sided)
• Ha: p > p0 Ha: p < p0 (1-sided)
• Set up hypotheses before getting the data
• Test statistic:
p p0
ˆ p p0
ˆ
z  33
 p 0 (1  p 0 ) / n
pˆ

Lab: Test of Proportion
• Suppose a coin toss turns up 12 heads out of 20 trials. At .05
significance level, can one reject the null hypothesis that the
coin toss is fair?
• Suppose that you interview 1000 exiting voters about who
they voted for PM. Of the 1000 voters, 550 reported that they

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voted for Rahul Gandhi. Is there sufficient evidence to suggest
that Rahul Gandhi will win the election at the .01 level?

34

Chi square test for Independence
• Chi square test of independence
• Is happiness independent of family income? A sample of 2955
families are studied

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Happiness
Very Pretty Not too Total
Income Above 272 294 49 615
Average 454 835 131 1420
Below 185 527 208 920

35

Chi-square statistic
• Summarize closeness of {fo} and {fe} by

( fo  fe )2
2  
fe

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where sum is taken over all cells in the table.

• When H0 is true, sampling distribution of this statistic is
approximately (for large n) the chi-squared probability
distribution.

36

Chi-square calculation
• In happiness and family income

( f o  f e )2 (272  189.6)2
 
2
  ...  172.3
fe 189.6

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• df = (3 – 1)(3 – 1) = 4. P-value = 0.000 (rounded, often reported as P <
0.001). Chi-squared percentile values for various right-tail probabilities
are in table on text p. 594.
• There is very strong evidence against H0: independence (If H0 were
true, prob. would be < 0.001 of getting this large a 2 test statistic or
even larger).
• For significance level a = 0.05 (or a = 0.01 or a = 0.001), we reject H0
and conclude that an association exists between happiness and
income. 37

Lab Chi-square distribution
• Suppose that 125 children are shown three television commercials for
breakfast cereal and are asked to pick which they liked best. The results are
shown in table below. You would like to know if the choice of favorite
commercial was related to whether the child was a boy or a girl or if these
two variables are independent

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A B C Totals
Boys 30 29 16 75
Girls 12 33 5 50
Totals 42 62 21 125

• Suppose you conducted a drug trial on a group of animals and you
hypothesized that the animals receiving the drug would show increased
heart rates compared to those that did not receive the drug. You conduct
the study and collect the following data.
Heart Rate Increased No Heart Rate Increase Total
38
Treated 36 14 50
Not treated 30 25 55
Total 66 39 105

Further Reading
• Test of samples means for two populations
• Test of sample proportions for two populations
• Odds ration for test of association

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39

Venkat Reddy Konasani
Manager at Trendwise Analytics
venkat@TrendwiseAnalytics.com
21.venkat@gmail.com

Venkat Reddy
+91 9886 768879

40

Testing of hypothesis

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