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Z Test & T Test
ROLL# 007
 Sometimes measuring every single piece of
item is just not practical
 Statistical methods have been developed to
solve these problems
 Most practical way is to measure a sample of
the population
 Some methods test hypotheses by comparison
 Two most familiar statistical hypothesis tests
are :
 T-test
 Z-test
 Z-test and T-test are basically the same
 They compare between two means to
suggest whether both samples come from
the same population
 There are variations on the theme for the Ttest
 Having a sample and wish to compare it with
a known mean, single sample T-test is
 Both samples not independent and have
some common factor (geo location, before after), the paired sample T-test is applied
 Two variations on the two sample T-test:
 The first uses samples with unequal
 The second uses samples with equal
 Use a Z-Test when you know the mean (µ) of
the population we are comparing our sample
to and the standard deviation () of the
population we are comparing our sample to.
 Use T-test for dependant samples when
subjects tested are matched in some way or
use T-test for independent samples when
subjects are not matched.
The Z-test compares the mean from a research
sample to the mean of a population. Details
(μ, σ) of the population must be known.
The t-test compares the means from two
research samples. Used when the population
details (μ, σ) are unknown.
 A T-test is a statistical hypothesis test
 The test statistic follows a Student’s Tdistribution if the null hypothesis is true
 The T-statistic was introduced by W.S.
Gossett under the pen name “Student”
 T-test also referred to as the “Student T-test”
 T-test is most commonly used Statistical Data
Analysis procedure for hypothesis testing
 It is straightforward and easy to use
 It is flexible and adaptable to a broad range of
T-test is best applied when:
 Limited sample size (n < 30)
 Variables are approximately normally
 Variation of scores in the two groups is not
reliably different
 If the populations’ standard deviation is
 If the standard deviation is known, best to
use Z-test
Various T-tests and two most commonly applied
tests are :
 One-sample T-test : Used to compare a sample
mean with the known population mean.
 Paired-sample T-tests : Used to compare two
population means in the case of two samples
that are correlated. Paired sample t-test is
used in ‘before after’ studies, or when the
samples are the matched pairs, or the case is
a control study.
 Data sets should be independent from each
other except in the case of the paired-sample ttest
 Where n<30 the t-tests should be used
 The distributions should be normal for the
equal and unequal variance t-test
 The variances of the samples should be the
same for the equal variance t-test
 All individuals must be selected at random
from the population
 All individuals must have equal chance of
being selected
 Sample sizes should be as equal as possible
but some differences are allowed
Assumptions: Matched pair, normal distributions, same
variance and observations must be independent of each
Steps in the calculation:
1. Set up hypothesis: Two hypotheses
H0=Assumes that mean of two paired samples =
H1=Assumes that means of two paired samples 
2. Select the level of significance: Normally 5%
3. Calculate the parameter: t = d /  s2 / n ,
n-1 is df
4. Decision making: Compare calculated value (cv) with
table value (tv). If cv  tv, reject H0 , If cv  tv, accept H0 and
say that there is no significant mean difference between the
two paired samples in the paired sample t-test.
 The Z-test is also applied to compare sample
and population means to know if there’s a
significant difference between them.
 Z-tests always use:
 Normal distribution
 Ideally applied if the standard deviation is
Z-tests are often applied if :
 Other statistical tests like t-tests are applied in
 Incase of large samples (n > 30)
 When t-test is used in large samples, the t-test
becomes very similar to the Z-test
 Fluctuations that may occur in t-tests sample
variances, do not exist in Z-tests
 Data points should be independent from each
 Z-test is preferable when n is greater than 30
 The distributions should be normal if n is low,
if n>30 the distribution of the data does not
have to be normal
 The variances of the samples should be the
 All individuals must be selected at random
from the population
 All individuals must have equal chance of
being selected
 Sample sizes should be as equal as possible
but some differences are allowed
Question #1: Does the research sample come
from a population with a known mean?
Example: Does prenatal exposure to drugs affect
the birth weight of infants?
Question #2: Is the population mean really what it
is claimed to be?
Examples: Does this type of car really run 12 kpl?
Does this diet pill really let people lose an average
of 25 pounds in 6 weeks?
Research question: Do Dhaka College students differ in
IQ scores from the average college student of BD?
Data : National average,  = 114,  = 15, N=150, X = 117
Steps in Calculation:
1. Set null and alternative hypothesis:(From data)
 H0:  = 114 , mean of the population from which we
got our sample is equal to 114.
 H1:   114 , mean of the population from which we
got our sample is not equal to 114.
2. Select level of significance, generally 5%
3. State decision rules : If zobs < +1.96 or zobs > -1.96,
reject H0
4. Compute standard error of mean: x = /N = 1.225
5. Calculate z-value: z = X - µ / x = + 2.45
6. Compare observed z to decision rules, and make
decision to reject or not reject null.
 2.45 > 1.96, so reject H0. so, more likely that the
sample mean is from some other population.
 Statistically significant difference between
sample mean and the population mean.
7. If H0 rejected, compare sample mean, and make a
conclusion about the research question:
 Observed mean was statistically significantly
greater than the population mean we compared it
to 117 > 114.
 So, it can be concluded that Dhaka College
students have higher IQ test scores than the
average college students of BD.
 Z-test is a statistical hypothesis test that follows a
normal distribution while T-test follows a Student’s Tdistribution.
 A T-test is appropriate when handling small samples
(n<30) while a Z-test is appropriate when handling
moderate to large samples (n > 30).
 T-test is more adaptable than Z-test since Z-test will
often require certain conditions to be reliable.
Additionally, T-test has many methods that will suit any
 T-tests are more commonly used than Z-tests.
 Z-tests are preferred than T-tests when standard
deviations are known.