business maths

1. Hypothesis Testing
Researchers are interested in answering many types of questions. For example,
i. A scientist might want to know whether the earth is warming up.
ii. A physician might want to know whether a new medication will lower a
person’s blood pressure.
iii. An educator might wish to see whether a new teaching technique is better than
a traditional one.
iv. A retail merchant might want to know whether the public prefers a certain
color in a new line of fashion.
v. Automobile manufacturers are interested in determining whether seat belts
will reduce the severity of injuries caused by accidents.
These types of questions can be addressed through statistical hypothesis testing,
which is a decision-making process for evaluating claims about a population.
z-test for mean:
The z test is a statistical test for the mean of a population. It can be used when
𝑛 ≥ 30, or when the population is normally distributed and s is known.
The formula for the z test is
𝑧 =
𝑥
̅ − 𝜇
𝜎
√𝑛
where
𝑥̅ = sample mean

2. 𝜇 = hypothesized population mean
𝜎 = population standard deviation
n = sample size
When population is normally distributed and s is known then
𝑧 =
𝑥
̅ − 𝜇
𝑠
√𝑛
Steps involving in hypothesis testing:
Step 1 State the hypotheses and identify the claim.
Step 2 Find the critical value(s).
Step 3 Compute the test value.
Step 4 Make the decision to reject or not reject the null hypothesis.
Explanation of each step
Step 1 State the hypotheses and identify the claim.
There can be three types of this step.
Type I: Two tailed test
𝐻0: 𝜇 = 𝜇0
𝐻1: 𝜇 ≠ 𝜇0 [Two tailed]

3. Type II: Right tailed test
𝐻0: 𝜇 = 𝜇0
𝐻1: 𝜇 > 𝜇0 [Right tailed]
Type III: Left tailed test
𝐻0: 𝜇 = 𝜇0
𝐻1: 𝜇 < 𝜇0 [Left tailed]

4. Step 2 Find the critical value(s).
The critical value separates the rejection region from the acceptance region. The
symbol for critical value is C.V.
Finding the critical values depends on value of 𝛼 and which tailed test it is.
We are going to discuss again three types to find critical value.

5. Type I: Finding critical value for two tailed test.
Let us suppose that 𝛼 is given as 0.01. Calculate 1 − 𝛼 = 0.99
Since it is two tailed test so divided the value of 1 − 𝛼 by 2
1 − 𝛼
2
=
0.99
2
= 0.495
In z-table find the value that is closest to 0.495 which is .4951. critical value will be 2.58 as
explained below. Write this value on the boundary of regection region as shown in fig above.

6. Type II: Finding critical value for Right tailed test.
Let us suppose that 𝛼 is given as 0.05. Calculate 1 − 𝛼 = 0.95
Since it is right tailed test so subtract 0.5 (half area) from the value of 1 − 𝛼
(1 − 𝛼) − 0.5 = 0.95 − 0.5 = .45
In z-table find the value that is closest to 0.45, which is .4505. critical value will be 1.65 as
explained below. Write this value on the boundary of regection region as shown in fig above.

7. Type III: Finding critical value for Left tailed test.
Let us suppose that 𝛼 is given as 0.05. Calculate 1 − 𝛼 = 0.95
Since it is right tailed test so subtract 0.5 (half area) from the value of 1 − 𝛼
(1 − 𝛼) − 0.5 = 0.95 − 0.5 = .45
In z-table find the value that is closest to 0.45, which is .4505. critical value will be 1.65 as
explained below. Write this value on the boundary of regection region as shown in fig above.

8. Step 3 Compute the test value.
Put the values in formula for z-test
𝑧 =
𝑥
̅ − 𝜇
𝜎
√𝑛
and compute z.
Step 4 Make the decision to reject or not reject the null hypothesis.
Check either the value calculated in step 3 lies in acceptance region (clean region)
or in the rejection region (shaded region).
a) If z lies in acceptance region then accept 𝐻0
b) If z lies in rejection region then reject 𝐻0

9. Example: (Right tailed test)
A researcher reports that the average salary of assistant professors is more than
$42,000. A sample of 30 assistant professors has a mean salary of $43,260. At
𝛼 = 0.05, test the claim that assistant professors earn more than $42,000 per year.
The standard deviation of the population is $5230.
Solution:
Step 1 State the hypotheses and identify the claim.
𝐻0: 𝜇 = 42000
𝐻1: 𝜇 > 42000 [Right tailed] (This is claim of researcher)
Step 2 Find the critical value(s).
As 𝛼 is given as 0.05. Calculate 1 − 𝛼 = 0.95
Since it is right tailed test so subtract 0.5 (half area) from the value of (1 − 𝛼)
(1 − 𝛼) − 0.5 = 0.95 − 0.5 = .45
In z-table find the value that is closest to 0.45, which is .4505. critical value will be
1.65

10. Step 3 Compute the test value.
𝑛 = 30, 𝑥̅ = 43260, 𝜎 = 5230
𝑧 =
𝑥
̅ − 𝜇
𝜎
√𝑛
=
43260 − 42000
5230
√30
= 1.32
Step 4 Make the decision to reject or not reject the null hypothesis.
Since 1.32 lies in the acceptance region so accept 𝐻0. Further claim is rejected.

11. Example: (Left tailed test)
A researcher claims that the average cost of men’s athletic shoes is less than $80.
He selects a random sample of 36 pairs of shoes from a catalog and finds the
following costs (in dollars).
𝑥̅ = 75 , 𝜎 = 19.2
Test the claim at 𝛼 = 0.10 level of significance.
Solution:
Step 1 State the hypotheses and identify the claim.
𝐻0: 𝜇 = 80
𝐻1: 𝜇 < 80 [Left tailed] (claim)
Step 2 Find the critical value(s).
As 𝛼 is given as 0.10. Calculate 1 − 𝛼 = 0.90
Since it is left tailed test so subtract 0.5 (half area) from the value of (1 − 𝛼)
(1 − 𝛼) − 0.5 = 0.90 − 0.5 = .4 . In z-table find the value that is closest to 0.4,
which is 0.4015. critical value will be 1.29

12. Step 3 Compute the test value.
𝑛 = 36, 𝑥̅ = 75, 𝜎 = 19.2
𝑧 =
𝑥
̅ − 𝜇
𝜎
√𝑛
=
75 − 80
19.2
√36
= −1.56
Step 4 Make the decision to reject or not reject the null hypothesis.
Since −1.56 lies in the rejection region so reject 𝐻0. Further claim is accepted.

13. Example: (Two tailed test)
The Medical Rehabilitation Education Foundation reports that the average cost of
rehabilitation for stroke victims is $24,672. To see if the average cost of
rehabilitation is different at a particular hospital, a researcher selects a random
sample of 35 stroke victims at the hospital and finds that the average cost of their
rehabilitation is $25,226. The standard deviation of the population is $3251. At
𝛼 = 0.01, can it be concluded that the average cost of stroke rehabilitation at a
particular hospital is different from $24,672?
Solution:
Step 1 State the hypotheses and identify the claim.
𝐻0: 𝜇 = 24672
𝐻1: 𝜇 ≠ 24672 [Two tailed] (claim)

14. Step 2 Find the critical value(s).
As 𝛼 is given as 0.01. Calculate 1 − 𝛼 = 0.99
Since it is two tailed test so divided the value of (1 − 𝛼) by 2
1 − 𝛼
2
=
0.99
2
= 0.495
In z-table find the value that is closest to 0.495 which is .4951. critical value will
be 2.58 .

15. Step 3 Compute the test value.
𝑛 = 35, 𝑥̅ = 25226, 𝜎 = 3251
𝑧 =
𝑥
̅ − 𝜇
𝜎
√𝑛
=
25226 − 24672
3251
√35
= 1.01
Step 4 Make the decision to reject or not reject the null hypothesis.
Since 1.01 lies in the acceptance region so accept 𝐻0. Further claim is rejected.