Z-Test with Examples

1. Z-TEST
BY
GROUP 04
B.Sc. (Hons.) Agriculture
2nd Semester
Assignment presented as the partial fulfillment of the requirement of
Course STAT-102
College of Agriculture
BZU, Bahadur Sub-Campus Layyah

2. DEFINATION
Z test is a statistical procedure used to test an
alternative hypothesis against a null
hypothesis.
Z-test is any statistical hypothesis used to
determine whether two samples’ means are
different when variances are known and
sample is large (n ≥ 30).
It is Comparison of the means of two
independent groups of samples, taken from
one populations with known variance.

3. The null hypothesis (H0) is
a hypothesis which the researcher tries to
disprove, reject or nullify. The 'null' often
refers to the common view of something,
while the alternative hypothesis is what the
researcher really thinks is the cause of a
phenomenon.

4. Z-TEST
Formula to find the value of Z (z-test) Is:
x̄ = mean of sample
μ0 = mean of population
σ = standard deviation of population
n = no. of observations

5. WHY WE USE LARGE SAMPLE…?
When we perform a statistical test
we are trying to judge the validity of
the null hypothesis. We are doing so
with an incomplete view of the
population. Our sample is our
window into the population.
The larger the sample size the bigger our window.
However without a full view of the population
there is always the chance that our sample will
lead us to the wrong conclusion.

6. When do we use Z-Test
When samples are drawn at random.
When the samples are taken from population
are independent.
When standard deviation is known.
When no. of observation is large (n ≥ 30)

7. EXAMPL
E

8. EXAMPLE
A principal at a school claims that the students
in his school are above average intelligence.
A random sample of thirty students’ IQ scores
have a mean score of 112.5. Is there sufficient
evidence to support the principal’s claim?
The mean population IQ is 100 with a standard
deviation of 15.•

9. EXAMPLE
• Step 1: State the Null hypothesis.
The accepted fact is that the population mean is
100, so: H0: μ=100.
• Step 2: State the Alternate Hypothesis.
The claim is that the students have above
average IQ scores, so:
H1: μ > 100.

10. EXAMPLE
• Step 3: Draw a picture to help you
visualize the problem.

11. EXAMPLE
• Step 4: State the alpha level.
If you aren’t given an alpha
level, use 0.05, An alpha level
of 0.05 is equal to a z-score
of 1.645.
To calculate z-score use
TI-83 calculator.

12. EXAMPLE
• Step 5: Find the Z using this formula:
For this set of data:
Z= (112.5-100) / (15/√30) = 4.56
• Step 6: If Step 5 (4.56) is greater than Step 4
(1.645), reject the null hypothesis. If it’s less than
Step 4, you cannot reject the null hypothesis. In
this case, it is greater, so you can reject the null
and principal’s claim is right.

13. Z-TEST FOR TWO SAMPLES
• Requirements: Two normally distributed but
independent populations, σ is known
• Formula:
• where x1 and x2 and are the means of the two
samples, μ 1 – μ 2 is the hypothesized difference
between the population means , σ 1 and σ 2 are
the standard deviations of the two populations,
and n 1and n 2are the sizes of the two samples.

14. EXAMPLE
• The amount of a certain trace element in blood is
known to vary with a standard deviation of 14.1
ppm (parts per million) for male blood donors
and 9.5 ppm for female donors.
• Random samples of 75 male and 50 female
donors yield concentration means of 28 and 33
ppm, respectively. What is the likelihood that the
population means of concentrations of the
element are the same for men and women?

15. EXAMPLE
• Null hypothesis: H 0: μ 1 = μ 2
or H 0: μ 1 – μ 2= 0
• alternative hypothesis: H a : μ 1 ≠ μ 2
or: H a : μ 1 – μ 2≠ 0

16. EXAMPLE
• The computed z‐value is negative because the
(larger) mean for females was subtracted from
the (smaller) mean for males.
• But the order of the samples in this computation
is arbitrary— it could be in opposite order, in
which case z would be 2.37 instead of –2.37. An
extreme z‐score from alpha level (which is 5% =
1.654 (plus or minus) will lead to rejection of the
null hypothesis.

17. NO RELEGION IS HIGHER
THAN
HUMANITY…!
(ABDUL SATTAR EDHI)