The document discusses the t-distribution and how it differs from the z-distribution. The t-distribution is a type of normal distribution used for smaller sample sizes where the population variance is unknown. It is similar in shape to the z-distribution but has thicker tails. As the degrees of freedom increase, the t-distribution becomes more similar to the z-distribution. The t-distribution is symmetrical around 0 and used to find t-values and percentiles using a t-table when the sample size is less than 30.

2. @Groupn't
Module 9:
T-DISTRIBUTION AND
PERCENTILE
10/07/2022
GROUP 9

3. @Groupn't
Module 9.1:
UNDERSTANDING THE T-
DISTRIBUTION
10/07/2022
GROUP 9

4. IMPORTANT TERMS:
STANDARD DEVIATION-The standard deviation
is the average amount of variability in your
dataset. It tells you, on average, how far each
value lies from the mean.
SAMPLE MEAN
POPULATION MEAN
VARIANCE: The variance is a measure of
variability. It is calculated by taking the
average of squared deviations from the
mean.
In this discussion, the population variance is
also referred to as the "population standard
deviation"

5. ANOTHER DAY, ANOTHER TYPE OF
NORMAL DISTRIBUTION
THE Z-DISTRIBUTION IS A TYPE OF NORMAL
DISTRIBUTION THAT CAN BE USED FOR CASES
WITH SAMPLE SIZES 30 OR ABOVE (REFER TO
MODULE 8- CENTRAL LIMIT THEOREM)
TO SOLVE CASES WHERE THE SAMPLE SIZE IS
LESS THAN 30, A NEW TYPE OF NORMAL
DISTRIBUTION IS CREATED. THIS DISTRIBUTION IS
CALLED "T-DISTRIBUTION" OR "STUDENT'S T-
DISTRIBUTION".

6. WHAT IS T-DISTRIBUTION
The t-distribution, also known as
Student’s t-distribution, is a way of
describing data that follow a bell
curve when plotted on a graph,
with the greatest number of
observations close to the mean
and fewer observations in the
tails.
It is a type of normal distribution
used for smaller sample sizes,
where the population variance is
unknown.
FIGURE 1: GRAPH OF T-DISTRIBUTION
T-DISTRIBUTION IS
INVENTED BY
WILLIAM SEALEY
GOSSET (1908)

7. WHAT'S NEW?
FIGURE 2: T-DISTRIBUTION AND Z-DISTRIBUTION
COMPARISON(GRAPH)
The t-distribution is almost
identical to the z-distribution.
Observe that the graph between
the two is similar in shape and
symmetry. In relation to this, the t-
distribution becomes more
identical to the z-distribution as the
measure of degrees of freedom
gets higher(Refer to next page).
t distribution gives a lower
probability to the center and a
higher probability to the tails than
the standard normal distribution.

8. WHAT'S NEW?
FIGURE 2: T-DISTRIBUTION AND Z-DISTRIBUTION
COMPARISON(GRAPH)
The variance is always
greater than 1. To find the
variance, use the formula:
v
v-2
We can assume that the standard
deviation is also greater than 1
v=number of degrees of freedom
and it is equal or more than 2.

9. T-DISTRIBUTION GRAPH
BELL-SHAPED
SYMMETRICAL
SYMMETRICAL ABOUT 0
AS YOU PLOT THE GRAPH OF T-
DISTRIBUTION, YOU WILL
OBSERVE THAT IT IS:
IS THERE A SKEWED T-
DISTRIBUTION?
-YES, BUT IT'S A DIFFERENT
CASE (SEARCH SKEWED
GENERALIZED T-DISTRIBUTION)
BELL-SHAPED
SYMMETRICAL

10. T-DISTRIBUTION THEOREM
If x̅ and s are the mean and standard deviation, respectively, of a random sample
of size n taken from a normally distributed population with a mean μ, can be
standardized as
Where;
x̅ = sample mean
μ = population mean
s = sample standard deviation
n = sample size
t =
x̅ − μ
s
√n
Remember that
the formula
should be used
when the sample
size (n) is less than
30.

11. Example 1:
What is the t-value when
μ=42, x̅ = 43, s=6 and n=25?

12. Example 1:

13. Example 2:
MATH Corporation manufactures light bulbs. The CEO
claims that an average Acme light bulb lasts 400 days. A
researcher randomly selects 20 bulbs for testing. The
sampled bulbs last an average of 380 days, with a
standard deviation of 50 days. Find the t-value of the
given data.

14. Example 2:

15. T-DISTRIBUTION Z-DISTRIBUTION
Used when the sample size
(n) is less than 30.
Bell-shaped
Symmetrical
A bit short but thick
GRAPH:
FORMULA:
Used when the sample size (n)
is equal or more than 30.
Bell-shaped
Symmetrical
Long but thin
GRAPH:
FORMULA:
t =
x̅ − μ
s
√n
z =
X − μ
σ
z =
x̅ − μ
√n
σ

16. SUMMARY(PROPERTIES OF T-DISTRIBUTION)
THE T-DISTRIBUTION IS SYMMETRICAL AT ABOUT 0.
THE T-DISTRIBUTION IS BELL-SHAPED LIKE THE NORMAL CURVE BUT HAS HEAVIER
TAILS.
THE MEAN, MEDIAN, AND THE MODE OF THE T-DISTRIBUTION ARE ALL EQUAL TO ZERO,
THE VARIANCE IS ALWAYS GREATER THAN 1. IT IS EQUAL TO WHERE V IS THE
NUMBER OF DEGREES OF FREEDOM.
AS THE DEGREES OF FREEDOM INCREASE, THE T DISTRIBUTION CURVE LOOKS MORE
AND MORE LIKE THE NORMAL DISTRIBUTION.
THE STANDARD DEVIATION OF THE T-DISTRIBUTION VARIES WITH THE SAMPLE SIZE.
THE TOTAL AREA UNDER AT T-DISTRIBUTION CURVE IS 1 OR 100%.
v
v-2

17. @Groupn't
Module 9.2:
PERCENTILES USING T-
TABLE
10/07/2022
GROUP 9

18. IMPORTANT TERMS:
DEGREES OF FREEDOM: refers to the maximum number
of logically independent values, which are values that
have the freedom to vary, in the data sample
CONFIDENCE INTERVAL: Confidence intervals use t-
scores to calculate the upper and lower bounds of the
prediction interval.
STANDARD DEVIATION: The standard deviation is the
average amount of variability in your dataset. It tells
you, on average, how far each value lies from the
mean.
T-SCORE: the number of standard deviations away
from the mean of the t-distribution

19. HOW TO FIND THE T-SCORE
TO FIND THE T-SCORE OF THE CASE OR GIVEN SAMPLES, YOU HAVE TO FIND THE DEGREES
OF FREEDOM FIRST. TO FIND THE DEGREE OF FREEDOM, USE THE FORMULA:
n-1 n=SAMPLE SIZE
STEPS ON FINDING THE T-SCORE:
Compute for the degree of freedom
Change the percentile into a percentage to decimal
Subtract the decimal to 1 To know what the right tail area is.
Referring to the table. Look for the column of 0.05 and the row of the df.
Write your answer
1.
2.
3.
4.
5.

20. Example 1:
Mr. Sotto conducts a survey of 25 people for the
effectiveness of their new
medicine. He wants to know what is the 95th percentile
of his survey. Find the t-value.

21. Example 1:

22. @Groupn't
Module 9.2:
GROUP BATTLE
10/07/2022
GROUP 9

24. REMINDER: THERE'S 1 ITEM WHERE THE
QUESTION IS INTENTIONALLY WRONG.
IF YOU ENCOUNTER THAT ITEM, LEAVE
NO ANSWERS IN YOUR BOARD/ PAPER

25. QUESTION 1:
t =
x̅ − μ
s
√n
GIVEN:
μ=42, x̅ = 43, s=6 and n=45
Based on the given values, which formula is
applicable?
a. b.
z =
x̅ − μ
√n
σ

26. QUESTION 1:
ANSWER:
EXPLANATION: z-distribution is applicable
when n is equal or more than 30.
b.
z =
x̅ − μ
√n
σ

27. QUESTION 2: Mr. Dela Cruz conducts a survey of 25 people for the
effectiveness of their new software. To determine the
degrees of freedom, he used the equation:
n-1
QUESTION: Did Mr. Dela Cruz use the correct equation for
finding the degrees of freedom?
a. Yes
b. No, the equation
should be n+3
c. No, the equation
should be n-5
d. the question is
canja baktol

28. QUESTION 2:
ANSWER:
EXPLANATION: Refer to page 19
a. Yes

29. QUESTION 3: When is T Distribution used?
a. When sample size(n) is less than 15 and the
population variance is known
b. When sample size(n) is equal to 30 and the
variance is unknown
c. When sample size(n) is less than 30 and the
population variance is unknown
d. when the sample size is unknown and the
population variance is less than 30

30. QUESTION 3:
ANSWER:
EXPLANATION: Refer to page 6
c. When sample size(n)
is less than 30 and the
population variance is
unknown

31. QUESTION 4:
What is the other term for T distribution?

32. QUESTION 4:
ANSWER:
EXPLANATION: Refer to page 6
Student's t-
distribution

33. QUESTION 5:
t-distribution becomes more identical to
the z-distribution as the measure of
degrees of freedom gets higher
Determine if the statement is true or
false

34. QUESTION 5:
ANSWER:
EXPLANATION:
TRUE

35. QUESTION 6:
If the graph of t-distribution is thicker and
shorter compared to z distribution, it
means that it gives _____ probability to the
center and _______ probability to the tails.

36. QUESTION 6:
ANSWER:
If the graph of t-distribution is thicker and
shorter compared to z distribution, it
means that it gives _____ probability to the
center and _______ probability to the tails.
lower
higher

37. QUESTION 7:
GIVEN:
μ=77 , x̅ =82 , s= 12 n= 22
Solve using the t-distribution theorem

38. QUESTION 7:
ANSWER:
1.95

39. QUESTION 8:
GIVEN:
μ=69 , x̅ =77 , s= 12 n= 99
Solve using the t-distribution theorem

40. QUESTION 8:
ANSWER:
INVALID
QUESTION
EXPLANATION: n=99 makes the given
unapplicable with the t-distribution theorem

41. QUESTION 9:
In t-distribution, the mean,
median, and mode are all
equal to:

42. QUESTION 9:
ANSWER:
0 or zero

43. QUESTION 10: Mr. Dela Cruz conducts a survey of 25 people for the
effectiveness of their new software. He wants to know
what is the 95th percentile of his survey. To do this, he
has to find the t-value. First, he determined the
degrees of freedom where he used the equation n-1.
After this, he got a df of 24.
QUESTION: What should he do next?
a. Recompute the degree of
freedom due to self-trust issues
b.Change the percentile into a
percentage to decimal
c. Look for the column of 24 and the
row of the percentile
d. That's it, right?

44. QUESTION 10:
ANSWER:
b.Change the
percentile into a
percentage to decimal

45. AND, THAT'S IT
THANK YOU FOR LISTENING!
HOPE YOU TOOK NOTES
GROUP 9