Z test
DR JATIN CHHAYA
ASSISTANT PROFESSOR
DEPARTMENT OF COMMUNITY MEDICINE,
GMERS MEDICAL COLLEGE, JUNAGADH
Learning Objectives
⚫ Understand Hypothesis
⚫ Level of Significance
⚫ Standard Error
⚫ Significance Level with Normal Distribution Curve
⚫ Process of Hypothesis Testing, Z test and
Interpretation of Results
⚫ Calculation- Difference between Two Mean or Two
Proportion
What we Going to Understand..
1. In a Study on Growth of Children, One group of 100
Children had a mean Height of 60 cm and SD of 2.5 cm
while another group of 150 children had a mean height of
62 cm and SD of 3 cm. Is the Difference between the two
group statistically Significant?
2. In a nutritional study, 100 children were given usual diet
and 100 were given vitamin A and D tablets. After 6
months, average weight of group A was 29kg with SD of
1.8kg and average weight of group B was 30kg with SD of
2kg. Is the difference is significant?
Introduction - Hypothesis
⚫ Hypothesis is a predictive statement, capable of
being tested by scientific methods, that relates an
independent variables to some dependent variable.
e.g.
⚫ Students who receive counseling will show a greater
increase in creativity than students not receiving
counseling
Characteristics of Hypothesis
⚫ Clear and precise.
⚫ Capable of being tested.
⚫ Stated relationship between variables.
⚫ Must be specific.
⚫ Stated as far as possible in most simple terms so that the same is easily
understand by all concerned.
⚫ Consistent with most known facts.
⚫ Responsive to testing with in a reasonable time. One can’t spend a life time
collecting data to test it.
⚫ Explain what it claims to explain; it should have empirical reference
Null Hypothesis
⚫ It is a Hypothesisi of No Difference (Nullifies the claim that Experimental
results is different from observed one) and we hold as true unless we have
sufficient statistical evidence to conclude otherwise.
⚫ Null Hypothesis is denoted by 𝐻0
⚫ If a population mean is equal to hypothesized mean then Null Hypothesis
can be written as
𝐻0:𝜇=𝜇0
Eg:
⚫ ‘there is no difference in the location of superstores and small grocers
shops’
⚫ ‘there is no connection between the size of farm and the type of farm’
Alternative hypothesis
⚫ The Alternative hypothesis is reversal of null hypothesis
and is denoted by 𝐻𝑎
⚫ If Null is given as - 𝐻0:𝜇=𝜇0
⚫ Then alternative Hypothesis can be written as
𝐻𝑎:𝜇≠𝜇0
𝐻𝑎:𝜇>𝜇0
𝐻𝑎:𝜇<𝜇0
Level of significance
⚫ Significance means the percentage risk to reject a
null hypothesis when it is true and it is denoted by 𝛼.
Generally taken as 1%, 5%, 10%
⚫ It simply means probability of committing
acceptable Type 1 error in statistical Test
Level Of Significance Z test Table Value
0.01 2.58
0.05 1.96
SE
⚫ Individual Variation: How Spread Out a data Set is
measured by SD
⚫ Group variation: How Spread Out a Data Set Measured
By SE
Eg: When we withdrawn a different samples from a Large
Population; obviously it have different sample means for
respective samples and Like SD; SE measures how this
means “Spread Out” around actual population mean.
SE
⚫ Like SD; SE also Follows the Normal Distribution
Curve, it means
1. Mean ± 1 SE limits include 68% of Sample Values
2. Mean ± 2 SE limits include 95% of Sample Values
3. Mean ± 3 SE limits include 99% of Sample Values
⚫ SE Does not means you made a mistake, its just a
Chance variation at Given Level of Significance
Significance level (Two Tailed Test)
A Sample Distribution Showing Rejection of H0 at LOS 5% and
Power 95%
Procedure for Hypothesis Testing
State the null (Ho)and alternate (Ha) Hypothesis
State a significance level; 1%, 5%, 10% etc.
Decide a test statistics; z-test, t-test, F-test.
Compute or Calculate Z value
If Calculate Z Value is greater than the Table Z value at Given
Level of Significance, that means it Reject the Null
Hypothesis (Difference is Significant)
And If Calculate Z Value is Smaller than the Table Z value at
Given Level of Significance, that means it Accept the Null
Hypothesis (Difference is Insignificant)
Calculation
DETERMINE APPROPRIATE TEST
Z TEST FOR THE MEAN
(STANDARD ERROR OF DIFFERENCE BETWEEN TWO MEANS)
Z TEST FOR THE PROPORTION
(STANDARD ERROR OF DIFFERENCE BETWEEN TWO PROPORTIONS)
TEST FOR SINGLE MEAN
Test for Single Mean
⚫ Suppose Government has Received a Complaint
Against municipal schools that boy of them were
underfed…
⚫ (In universe – from literature) Average weight of
Boys of age 10, is 32 Kgs with S.D= 9 Kgs
⚫ A sample of 100 boys were selected from municipal
school of same age and average weight found to be
29.5 kg
Test for Single Mean
⚫ At α =0.05, we need to check whether the complaint
is true or not (we check the average weight is
closure to mean of population or far away
from population)
⚫ H0 : 𝜇 = 32, no significant difference between two
students
⚫ H1 : 𝜇 < 32, there is significant difference between
students and complaint is true
⚫ If we accept H1; than what it mean Complain is true
Test for Single Mean
⚫ Z = X1 - 𝜇 As obtained value of z (-2.77) is
SE of Mean higher than critical value (Z
⚫ Z = X1 - 𝜇 Table value at LOS 5% is 1.96),
σ/ √n
X1 = Sample mean The observed difference is
𝜇 = Population Mean
σ = Standard Deviation of Population significant, we Reject the Null
N = number of Sample Hypothesis and accept the
alternate hypothesis and
Z = 29.5 – 32
conclude that students of
9/ √100
Municipal school were underfed
Z = - 2.77
and underweight.
Calculation: Standard Error of Difference B/w Two Means of
Large Sample
⚫ Null Hypothesis Explain That if Samples are drawn
Randomly and Sufficient in Large Size, their mean should
not differ from Population Mean.
⚫ Non-Zero Value of Difference between Two Means is only
acceptable when
“Calculated Z value” is smaller than “table Z value” (means lesser than 1.96
times of SE at LOS 5%) and we accept the Null Hypothesis
“Calculated Z value” is Greater than “table Z value” (means greater than 1.96
times of SE at LOS 5%) and we Reject the Null Hypothesis
STANDARD ERROR OF DIFFERENCE BETWEEN TWO MEANS
X1 =Mean for Sample 1
X2 = Mean for Sample 2
SD1 = Standard deviation for
Sample 1
SD2 = Standard deviation for
Sample 2
N1 – Sample Size for Sample 1
N2 – Sample Size for Sample 2
EXAMPLE
⚫ In a nutritional study, 100 children were given usual diet and 100 were
given vitamin A and D tablets. After 6 months, average weight of group
A was 29kg with SD of 1.8kg and average weight of group B was 30kg
with SD of 2kg. Is the difference is significant?
SD1= 1.8 n1 = 100
SD2= 2 n2 = 100
⚫ Ho: There is No Significant Difference between mean
weight in both of the group
⚫ H1: The difference observed in Mean is Significant
⚫ LOS : 0.05
As obtained value of z (3.7) is higher
than critical value (Z Table value at
LOS 5% is 1.96),
The observed difference is significant,
we Reject the Null Hypothesis and
accept the alternate hypothesis and
conclude that vitamins played a role in
weight gain
TEST FOR PROPORTIONS
Test for Single Proportion
⚫ In a medical college, out of 120 admission of 1st
MBBS, 35 are girl students. Check whether the
proportion of girl students is 40% in this college.
p= proportion of success
⚫ Z=p–P P = proportion of success in universal population
q= 100-p
SE (p) n = sample size
⚫ Z=p–P
√pq/n p= 35*100/120 =29%
P = 40%
q = 100-p= 100-29 = 71%
⚫ Z=p–P As obtained value of z (-2.65) is
higher than critical value (Z
√pq/n
Table value at LOS 5% is 1.96),
Z = 29 -40
√ 29 * 71/120 The observed difference is
significant, we Reject the Null
Z= 11/ √ 17.15 Hypothesis and accept the
Z = 11/ 4.14 alternate hypothesis and
conclude that the proportion of
Z = -2.65
girls student is not 40%
STANDARD ERROR OF DIFFERENCE
BETWEEN TWO PROPORTIONS
P1: Probability of Success in
Sample 1
P2: Probability of Success in
Sample 2
Q1 = 100 – P1
Q2 = 100 – P2
N1 – Sample Size for Sample 1
N2 – Sample Size for Sample 2
EXAMPLE
⚫ If swine flu mortality in one sample of 100 is 20%
and in another sample of 100 it is 30%. Is the
difference in mortality rate is significant?
P1= 20 Q1 = 80 n1 = 100
P2= 30 Q2 = 70 n2 = 100
⚫ Ho: There is No Significant Difference between
Proportion of Mortality both of the Sample
⚫ H1: The difference observed in Proportion of
Mortality is Significant
⚫ LOS : 0.05
As obtained value of z (1.64) is
lesser than critical value (Z
Table value at LOS 5% is 1.96),
The observed difference is
insignificant, we Accept the
Null Hypothesis and Reject the
alternate hypothesis and
conclude that Observed
Difference in Mortality is only
by Chance.
Example
⚫ In a School A, Tonsillectomy had been done in 23 students out of 50
while in the other School B it was done in 44 Students out of 200. Is
this difference is Statistically Significant?
⚫ Ho: There is No Significant Difference between Proportion of
Tonsillectomy in both of the Sample
⚫ H1: The difference observed in Proportion of Tonsillectomy is
Significant
⚫ LOS : 0.05
⚫ School A:
n1= 50
P1 (probability of Succeess)= 23/50 *100 = 46%
Q1 = 100- 46= 54%
⚫ School B:
n2= 200
P2 (probability of Succeess)= 44/200 *100 = 22%
Q2 = 100- 46= 88%
Calculate Z= 3.11
As obtained value of z (3.11) is
higher than critical Z value (Z
Table value at LOS 5% is 1.96),
The observed difference is
significant, we Reject the Null
Hypothesis and accept the
alternate hypothesis and
conclude difference observed
occurrence of Tonsillectomy is
significant.
Home Work..
1. In a Study on Growth of Children, One group of 100 Children had a
mean Height of 60 cm and SD of 2.5 cm while another group of 150
children had a mean height of 62 cm and SD of 3 cm. Is the Difference
between the two group statistically Significant?
2. Find the significant of difference in the Mean heights of 50 girls and
50 Boys with the following Values..
Mean SD SE
Girls 147.4 Cm 6.6 cm 0.93
Boys 151.6 Cm 6.3 cm 0.89
Example..
⚫ In a School A, there was a History of Whooping Cough in 25
students out of 50 while in the other School B History of
Whooping Cough in 88 Students out of 200. Determine
Whether the Difference is Due to Chance or Real.
Thank you…
