1. Presentation on Data Analysis
Z Test & T Test
PRESENTED BY : GROUP 1
MD SHAHIDUR RAHMAN ROLL# 003
MD AMINUL ISLAM ROLL# 007
MRS ROZINA KHANAM ROLL# 038

2. Introduction
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
Cont’d

3. Introduction 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 T-test
Having a sample and wish to compare it with a known mean, single sample T-test is applied
Cont’d

4. Introduction
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 variances
The second uses samples with equal variances

5. When to use Z-Test and T-Test
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.

6. Comparing Z-test and t-test
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.

7. T-test A T-test is a statistical hypothesis test
The test statistic follows a Student’s T-distribution 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 circumstances
Cont’d

8. T-test T-test is best applied when: Limited sample size (n < 30)
Variables are approximately normally distributed
Variation of scores in the two groups is not reliably different
If the populations’ standard deviation is unknown
If the standard deviation is known, best to use Z-test
Cont’d

9. T-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.

10. Data types can be analysed with T-tests
Data sets should be independent from each other except in the case of the paired-sample t-test
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
Cont’d

11. Data types can be analysed with T-tests
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

12. Sequence of T-Test (Paired Sample)
Assumptions: Matched pair, normal distributions, same variance and observations must be independent of each other.
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.

13. Z-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 known
Cont’d

14. Z-Test Z-tests are often applied if :
Other statistical tests like t-tests are applied in substitute
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

15. Data types can be analysed with Z-test
Data points should be independent from each other
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 same
Cont’d

16. Data types can be analysed with Z-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

17. Z-test may ask two questions:
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?

18. Sequence of Z-Test (One-Sample)
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 < or zobs > -1.96, reject H0
4.   Compute standard error of mean: x = /N = 1.225
5.   Calculate z-value: z = X - µ / x = Cont’d

19. Sequence of Z-Test (One-Sample)
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.

20. Summary
Z-test is a statistical hypothesis test that follows a normal distribution while T-test follows a Student’s T-distribution.
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 need.
T-tests are more commonly used than Z-tests.
Z-tests are preferred than T-tests when standard deviations are known.

21. Q & A