1. Business Statistics Topic 7
Hypothesis Testing

2. Business Statistics:Topic 7
Learning Objectives
By the end of this topic you will be able to:
define the principle of statistical inference
formulate & distinguish the types of hypotheses
recognise different types of errors in hypothesis testing
recognise decision making using ‘Z’ test, ‘t’ test and ‘p’ value.
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3. Business Statistics:Topic 7
Decision Making
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4. Hypothesis testing Process
Population
Evidence to
accept our claim
We believe the average mileage is 10km (hypothesis)
Sample mean,
is 11 , test
Draw Sample
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5. What is a Hypothesis?
A Statement(s) or claim about a population parameter, () developed for the purpose of testing.
Hypothesis statement(s) are made before analysis.
We claim that the average mileage is 10km/l
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6. Business Statistics:Topic 7
Null Hypothesis
A Null Hypothesis is a statement that nothing unusual occurs or will occur.
We begin with the assumption it is true
Designated H0
Always has equality sign: =
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7. Alternative Hypothesis
Opposite of null hypothesis, H0
Alternative hypothesis is a statement that something unusual occurs or will occur:
Designated H1 or HA
Always has inequality sign: ¹, <, or >
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8. Business Statistics:Topic 7
Example 1
Test that the population mean is not 5
State the question statistically: m ¹ 5
State the opposite statistically: m = 5
Must be mutually exclusive & exhaustive
Select the null hypothesis: m = 5
Always has an “=“ sign
Hence the alternate hypothesis is: m ¹ 5 (two tailed test)
Never has an “=“ sign.
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9. Business Statistics:Topic 7
Example 2
Test that the population mean is less than 5
State the question statistically: m < 5
State the opposite statistically: m  5
Must be mutually exclusive & exhaustive
Select the null hypothesis: m  5
Always has an “=“ sign
Hence the alternate hypothesis is: m < 5 (one tailed test), Never has an “=“ sign.
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10. Business Statistics:Topic 7
One or Two- tailed Test?
Two-tailed test
The alternative hypothesis, H1 has a “¹ “ sign
One-tailed test
The alternative hypothesis, H1 has a “>” sign (right tail test) or
The alternative hypothesis, H1 has a “<“ sign (left tail test)
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11. Business Statistics:Topic 7
Type of Test : z or t ?
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12. Business Statistics:Topic 7
Level of Significance
Probability of rejecting a true null hypothesis
Represented by ‘a’ (alpha)
Selected by researcher at start
Typical values are .01, .05, .10
Same as confidence level of (1-a) 100%
For example, a = .01 means 99% confidence level
Used for getting the critical value stating the decision rule
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13. Critical Value & Rejection Region
The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.
based on selected  value
and the type of test (z or t)
and whether a one or two-tailed test is used.
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14. Critical Value & Rejection Region
H0: m = 5
H1: m ¹ 5
For a two-tailed test (H1:   5) both the upper and lower tails marked by the critical values are the rejection region for H0
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15. Critical Value & Rejection Region
H0: m = 5
H1: m > 5
For a one-tailed (right) test (H1:  >5) the rejection region is the right-hand tail marked by a positive critical value.
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16. Critical Value & Rejection Region
H0: m = 5
H1: m < 5
For a one-tailed (left) test (H1:  < 5) the rejection region is the left-hand tail marked by a negative critical value.
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17. Business Statistics:Topic 7
Decision Rule
If the calculated test statistic lies in the rejection
region, the null hypothesis is rejected
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18. Business Statistics:Topic 7
Assumptions
The population follows a normal distribution or
The sample size is sufficiently large.
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19. Steps in Hypothesis Testing
Set up the null and alternative hypothesis H0, H1 based on the research question
Decide the test to perform (z or t) depending on whether  is know or unknown
State the decision rule
Compute the test statistic (z or t) depending on the type of test used
Make decision using the rule (reject / accept H0)
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20. Z Test of Hypothesis for 
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21. Business Statistics:Topic 7
Test Statistic – Z test
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22. Critical Value & Decision Rule
Two-tailed test
When a = .05,
95% confidence level
Critical values are 1.96
(from ‘z’ table)
Note that for a two-tailed test the rejection region is on both tails
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23. Critical Value & Decision Rule
One-tailed (left) test
When a = .05,
95% confidence level
Critical value is
(from ‘z’ table)
Note that for a one-tailed test the rejection region is on only one side: left tail
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24. Critical Value & Decision Rule
One-tailed (right) test
When a = .01,
99% confidence level
Critical value is 2.33
(from ‘z’ table)
Note that for a one-tailed test the rejection region is on only one side: right tail
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25. Business Statistics:Topic 7
Example
The quality control manager believes that the lifetime of bulbs follows a normal distribution with a mean of 200 hours and a standard deviation of 20 hours.
A sample of 100 bulbs showed a sample mean of 195 hours. The manager wants to test whether the mean lifetime of all bulbs is 200 at a 1% level of significance.
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26. Business Statistics:Topic 7
Solution
Is the mean life time of bulbs 200 hrs?
Step 1
H0.:  = 200
H1:   200
(two-tailed test)
Step 2
Z or t test?
 known : Z test
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27. Business Statistics:Topic 7
Step 3
a is 1% = .01 each tail area is a /2 = .005
critical value is 2.58
(Use Z table)
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28. Business Statistics:Topic 7
Step 4
Step 5
Since the test statistic value, z = -2.5 does not lie in the
rejection area, we do not reject H0.
We conclude that the average lifetime of batteries is
200 hours.
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29. t-Test of Hypothesis for 
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30. Business Statistics:Topic 7
t-Test Statistic
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31. Business Statistics:Topic 7
Example
A particular branch of a well-known bank stores enough money during the weekends to satisfy its customer’s needs. The expected average withdrawal during the weekend is $500. When they looked at a sample of the last 16 weekend transactions, they found the average withdrawal to be $540 with a standard deviation of $70. At  = .05, is there evidence that the mean withdrawal has increased during weekends? It is assumed that the population follows a normal distribution.
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32. Business Statistics:Topic 7
Solution
Is the mean withdrawal more than $500?
Step 1
H0.:   500
H1:  > 500
(one-tailed (right) test)
Step 2
Z or t test?
 unknown : t test
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33. Business Statistics:Topic 7
Step 3
a is 5% = .05 upper tail is .05
Degrees of freedom v = n-1 = 16-1=15
Critical value, tn-1 is (Use t table)
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34. Business Statistics:Topic 7
Step 4
Step 5
Since the test statistic value, t = lies in
the rejection area, we reject H0. We conclude
that the average money spent by the customers during
weekend has increased.
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35. Business Statistics:Topic 7
p-value test for 
The p value is the smallest value of the significance level at which H0 can be rejected
Assuming that the null hypothesis is true, the p value is the probability of obtaining a test statistic equal to or more extreme than the result obtained from the sample
If the p value is less than , the null hypothesis is rejected
If the p value is greater than or equal to , the null hypothesis is not rejected
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36. p-value for a two-tailed test
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37. p-value for a one-tailed(right) test
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38. p-value for a one-tailed(left) test
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39. Business Statistics:Topic 7
Example
Is the mean life time of bulbs 200 hrs?
Step 1
H0.:  = 200
H1:   200
(two-tailed test)
Step 2
 known :
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40. Business Statistics:Topic 7
Step 3
p-value = 2  Probability {Z  2.5} = 2  =
Step4
Since the p-value is less than the  = .05, we reject the
null hypothesis. Note that the decision is the same as
before.
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41. Business Statistics:Topic 7
Types of Error
In reality
H0 Is true
H0 Is
false
Decision
Type 1 error
Probability 
Reject H0
No error
Type 2 error
Probability 
Accept H0
No error
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42. Business Statistics:Topic 7
Types of Error
Type 1 error has serious consequences
As  increases  decreases
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43. Business Statistics:Topic 7
Summary
In this topic you have discussed:
Decision Making processes
Formulating and testing hypotheses
Hypothesis testing of  when  is known
Hypothesis testing of  when  is unknown
p-value approach in hypothesis testing
Types of errors involved in testing a hypothesis
Business Statistics:Topic 7