1. A Presentation on Statistical Hypothesis
Testing:
The t -test
Presented By:
POOJA DHAR
RISHIKESH DUTTA
SATABDI HAZARIKA
JYOTI PRASAD PEGU

2. What we are going to Discuss?
• What is a t-test?
• How to do an t-test?
• An Example t-test.
• A Practical representation of t-test in spread sheet.

3. Overview of Statistical Hypothesis Testing
Definitions
• Inferential Statistics: Procedures for deciding whether sample data represent a
particular relationship in a population. Used for deciding whether the sample
mean is representative of the population. a.k.a. hypothesis testing procedures
• Parametric Statistics: Inferential procedures that require assumptions about
the parameters of the raw score populations the data represent. They are
performed when it is appropriate to compute the mean.
• Nonparametric Statistics: Inferential procedures that do not require
assumptions about the parameters of the raw score population. Appropriate
when the median or the mode is the best measure of central tendency.

4. Overview of Statistical Hypothesis Testing
Definitions
• There are many different hypothesis testing procedures, using one
versus another will be based on the exact situation we find
ourselves in.
• Today we will learn the t-test.

5. occurred because of differences in our reading strategies, rather than by
chance?
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Suppose we conducted a study to compare two strategies for teaching spelling.
Group A had a mean score of 19. The range of scores was 16 to 22, and the standard deviation
was 1.5.Group B had a mean score of 20. The range of scores was 17 to 23, and the standard
deviation was 1.5.
How confident can we be that the difference we found between the means of Group A
and Group B
What is a t-test

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A t-test allows us to compare the means of two groups and determine how likely the difference
between the two means occurred by chance.
The calculations for a t-test requires three pieces of information:
- the difference between the means (mean difference)
- the standard deviation for each group
- and the number of subjects in each group.

7. All other factors being equal, large differences between means are less likely to
occur by chance than small differences.
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The size of the standard deviation also influences the outcome of a t-
test. Given the same difference in means, groups with smaller standard deviations
are more likely to report a significant difference than groups with larger standard
deviations.

9. Less overlap would indicate that the
groups are more different from each other.
than larger standard deviations.
From a practical standpoint, we can see that smaller standard deviations produce less overlap
between the groups
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The size of our sample is also important. The more subjects that are involved in a
study,
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the more confident we can be that the differences we find between our groups did not
occur by chance.

11. Once we calculate the outcome of the t-test (which produces a t-value), we
check that value (with the appropriate degrees of freedom) on a critical
value table to determine how likely the difference between the means
occurred by chance.
The above process can be accomplished with a computer statistical
package which calculates the means and standard deviations of both
groups, the mean difference, the standard error of the mean difference,
and a p-value (probability of the mean difference occurring by chance).
We have created an excel spreadsheet which does these calculations and
provides this information.

12. A correlated (or paired) t-test is concerned with the difference between the
average scores of a single sample of individuals who is assessed at two
different times (such as before treatment and after treatment) or on two
different measures. It can also compare average scores of samples of
individuals who are paired in some way (such as siblings, mothers and
daughters, persons who are matched in terms of a particular
characteristics).
An independent t-test compares the averages of two samples that are
selected independently of each other (the subjects in the two groups are
not the same people). There are two types of independent t-tests: equal
variance and unequal variance.
There are three types of t-tests and each is calculated slightly differently.

13. An equal variance (pooled variance) t-test is used when the number of subjects in the two
groups is the same OR the variance of the two groups is similar.
An unequal variance (separate variance) t-test is used when the number of subjects in the
two groups is different AND the variance of the two groups is different.

14. Are the scores for the two means from the
same subject (or related subjects)?
Paired t-test (Dependent t-test;
Correlated t-test)
Yes
Equal Variance Independent t-
test (Pooled Variance
Independent t-test)
Yes
Unequal Variance Independent
t-test (Separate Variance
Independent t-test)
Yes
(Significance Level for
Levene (or F-Max) is p <.05
Equal Variance Independent t-
test (Pooled Variance
Independent t-test)
No
(Significance Level for
Levene (or F-Max) is p >.05
Are there the same number of
people in the two groups?
No
Are the variances of the two
groups different?
No
How do we determine which t-
test to use…

15. How to do a Statistical Hypothesis Testing
• Hypothesis testing is an inferential statistical procedure that takes information
from a sample or two to evaluate hypothesis about a population.
• Often, in hypothesis testing, a treatment is added to a sample of the
population with a known mean which changes the mean of the sample in
some unknown way.
– If an effect occurs, it is to add or subtract a constant to the mean.
– As an example, perhaps we know the mean CAT score, however, we
implement a class to a sample of 10 students that is thought to change CAT
scores of the sample. The population of CAT scores has a known mean,
however the mean of the population of scores after the implementation of

16. Statistical Hypothesis Testing
Six Steps
Hypothesis testing involves a Six step procedure:
1. Stating the hypotheses.
2. Choose the level of significance.
3. Determine the critical values.
4. Calculate the test statistic.
5. Compare the test statistic to the critical values.
6. Write a conclusion.

17. Statistical Hypothesis Testing
Step 1
1. Stating the Hypotheses
•There are always two hypotheses formed when doing hypothesis testing. They
are called the null and alternative hypotheses.
•The null hypothesis (H0) states that nothing occurs. Implementing a treatment
does not change people’s scores. No difference between the means of the
treated and untreated populations.
•The alternative hypothesis (H1) states that the treatment does have an effect.
Implementing a treatment does change people’s scores. The mean of the treated
population is different from the mean of the untreated population.

18. Statistical Hypothesis Testing
Step 1
• In numerical terms:
–H0: µtreated = µuntreated
–H1:µtreated ≠ µuntreated
• Together, these two hypotheses must account for every possible thing that
could occur. In this case, either the means are equal or they are not equal.
Every possible outcome is covered.

19. Statistical Hypothesis Testing
Step 1
•Hypotheses can be directional or nondirectional.
•The hypotheses we’ve just written are nondirectional in that they
don’t predict what the treatment is going to do, increase or decrease
performance. These hypotheses will simply test whether the treated
and untreated means are different.
•If you have a reason to believe what effect the treatment is going to
have, you should use directional hypotheses.

20. Statistical Hypothesis Testing
Step 1
• For example, you’ve created a class that you think will improve CAT scores.
Your hypotheses will then read:
–H0: µtreated ≤ µuntreated
–H1: µtreated > µuntreated
• The null covers the possibilities that either nothing happened, or the opposite
of what I expected happened.
• Together these hypotheses cover every possible outcome that could occur.

21. Statistical Hypothesis Testing
Step 2
2. Choose the level of significance
•Choose the level of significance: a , the probability of making a Type I Error if H0
is true.
TRUE FALSE
ACCEPT Correct Decision TYPE 2 ERROR (β)
REJECT TYPE 1 ERROR (a) Correct Decision

22. Statistical Hypothesis Testing
Step 2
• The level of significance is often standardized to be 0.05, i.e. 5% in many cases
or 0.01, i.e. 1%.
• Which in turn means 95% or 99% of Confidence Interval.

23. Statistical Hypothesis Testing
Step 3
3. Determine the critical values.
•The critical value of t-statistics for the particular level of significance and degree of
freedom can be observed from the t-table.
• The Degree of Freedom is given as (n-1) where n= sample size
•Pretend that we were working on the CAT score problem. We want to know the t-scores
associated with an unlikely outcome. If we were told that α = .05 and df(degree of
freedom)=9 then we would need to find the t-score beyond which 5% of the distribution
lies. Looking in t-table , we find that at a t-score of +2.262, 5% of the scores fall in the tail.
•If we had been told that α = .01, then a t-score of + 3.250 would be needed.

24. Statistical Hypothesis Testing
Step 3
•If we were using nondirectional hypotheses, we would need to split the region
of rejection into both tails. That is, when using nondirectional hypotheses, an
extreme sample mean in either direction would be important.
•If we were using nondirectional hypotheses, and α was set at .05, then we
would have two critical t-score values. If the z-score from the sample was
greater than +2.262 or less than - 2.262 then it would fall into the most extreme
5%, 2.5% in each end.
•Likewise, if α=.01 for nondirectional hypotheses, then the critical t-scores would
be + 3.250 and - 3.250, these separate the most extreme 1%, ½ % in each tail.

25. Statistical Hypothesis Testing
Step 4
4. Calculate the test statistic.
•This step of hypothesis testing is pretty easy for now. We simply need
to figure out the mean of our treated sample and the standard error of
the sample. Once we have these, then we will plug them into the
t-score formula along with the original population mean.
•This is the step where the procedure will be different for each of the
different inferential procedures we have learned.

26. Statistical Hypothesis Testing
Step 4

27. Statistical Hypothesis Testing
Step 5 & 6
5. Compare the test statistic to the critical values:
•Using the information we have, we will evaluate the merits of our hypotheses.
•If tobt ≤ tcrit or tobt > tcrit
•Does the t-score from the treated sample fall in the region of rejection.
6. Write a conclusion.
– If so, then we would reject H0 and conclude that the treatment did have an
effect.
– If not, then we would fail to reject H0 and conclude that the evidence does
not support the idea that the treatment had an effect.

28. Statistical Hypothesis Testing
t-test Example

29. Statistical Hypothesis Testing
t-test Example
Step 1
State the Hypotheses
• H0: The puppy chow did not make the dogs grow any more than normal.
• H1: The puppy chow did make the dogs grow larger then normal
• H0: µchow ≤ µall
• H1: µchow > µall
• Notice that this is a one-tailed test, I predict that the Chow will make the dogs
grow, not shrink.

30. Statistical Hypothesis Testing
t-test Example
Step 2
Choose the level of significance
• A “very unlikely sample” in this case is one which occurs less than 5% of the
time by random chance.
• α = .05

31. Statistical Hypothesis Testing
t-test Example
Step 3
Determine the critical values
•Looking in t-table, we find that the t-score which sets off the top 5% of the
distribution with 9 degrees of freedom is .
•Therefore, tcrit = +2.262
•This value defines my region of rejection.
•If my tobt falls anywhere in this region of rejection, then we will reject H0.

32. Statistical Hypothesis Testing
t-test Example
Step 4
Calculate the test statistics

33. Statistical Hypothesis Testing
t-test Example
Step 5
Compare the test statistic
to the critical values
• tobt > tcrit
• The tobt falls in the region of rejection.
• We will reject H0.

34. Statistical Hypothesis Testing
t-test Example
Step 6
Write a conclusion
• Thus we conclude that the Super Vitamin
Enriched Puppy Chow makes Doberman
Pincers grow significantly larger.
• I can safely go ahead with my advertising
campaign!