Business statistics about hypothesis testing

1. 1
CHAPTER SEVEN
HYPOTHESIS TESTING
Introduction
In Chapter two, Estimation, we used the information obtained in a simple
random sample to construct a confidence interval estimate of the unknown value
of a population parameter. In this chapter, hypothesis testing, we start with an
assumed value of a population parameter: then we shall use sample evidence to
decide wither the assumed value is unreasonable and should be rejected or
whether it should be accepted.
The assumptions we make about the values of population parameters are called
hypotheses. Sample evidence is used to test the reasonableness of hypotheses;
hence, the statistical inferences made in this chapter are referred to as hypothesis
testing. A procedure based on sample evidence and probability theory to
determine whether the hypothesis is a reasonable statement is called hypothesis
testing.
In hypotheses testing we begin by making a tentative assumption about a
population parameter. This tentative assumption is called the null hypothesis,
and is denoted by Ho - it is the assumption we wish to test. We then define
another hypothesis, called the alternative hypothesis, which is the opposite of
what is stated in the null hypothesis. This alternative hypothesis specifies all
possible values of the population parameter that are not specified in the null
hypothesis, and in denoted by Ha. The hypothesis testing procedure involves
using data from a sample to test the two competing statements indicated by Ho
and Ha. Ho and Ha are mutually exclusive and collectively exhaustive.
In the process of hypothesis testing, the null hypothesis is initially assumed to be
true. The data are gathered and examined to determine whether the evidence is
strong enough away from the null hypothesis to reject it when the researcher in
testing an industry standard or a widely accepted values, the standard or
accepted value is assumed to be true in the null hypothesis. Null in this sense
means that nothing is new, or there in no new value or standard. The burden is
then placed on the researcher to demonstrate through gathered data that the null
hypothesis is false.
Ha, Hi = research hypothesis a statement that in accepted if the sample data
provide enough evidence that the Ho is false.

2. 2
The situation encountered in hypothesis testing is similar to the one encountered
in a criminal trial. In a criminal trial the assumption is that the defendant is
innocent. Thus, the null hypothesis is one of innocence.
The opposite of the null hypothesis is the alternative hypothesis that the
defendant is guilty. Thus, the hypothesis far a criminal trial would be written
Ho: The defendant is innocent
Ha: The defendant is guilty
To test these competing statements, or hypotheses, a trial is held. The testimony
and evidence obtained during the trial provide the sample information. If the
sample information is not inconsistent with the assumption of innocence, the null
hypothesis that the defendant is innocent can not be rejected. However, if the
sample information is inconsistent with the assumption of innocence, the null
hypothesis will be rejected. In this case, action will be taken based upon the
alternative hypothesis that the defendant is guilty.
Example
1. The manager of a hotel has stated that the mean guest bill for a weekend is Birr
400 or less. A member of the hotel’s accounting staff has noticed that the total
charges for guest bills have been increasing in recent months. The accountant
will use a sample of weekend guest bills to test the manager’s claim.
Required: State the null and alternative hypotheses
Solution:
Ho: μ  Birr 400
Ha:   Birr 400
2. Production workers at XY Company have been trained in their jobs by using two
different training programs. The company training director would like to know
whether there is a difference in mean productivity for workers trained in the two
programs.
Required: Develop the null and alternative hypotheses.
Solution
Ho: 1 = 2 or 1 - 2 = 0
Ha: 1  2 1 - 2  0
3. The manager at a drugstore claims that the company’s employees are honest.
However, there have been many shortages from the cash register lately.
Required: Specify the null and alternative hypothesis
Solution:
Ho: Employees are honest
Ha: Employees are dishonest
“In many situations, the choice of Ho and Ha is not obvious; in such cases,
judgment on the part of the user is needed to select the proper farm of Ho and

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Ha. However, the equality part of the expression (either =, or ) always appears
in the null hypothesis.
Type I and Type II Errors
There are four possible outcomes of any hypothesis test, two of which are correct
and two of which are incorrect. The incorrect ones are called type I and Type II
errors.
Type I Error
In hypothesis testing sample evidence is used to test the null hypothesis Ho.
Occasionally the sample data gathered in research process lead to a decision to
reject a null hypothesis when actually it is true. A type I error is committed
when a true null hypothesis is rejected. In short, rejecting a true Ho is called
Type I error. The possibility of committing a Type I error is represented by
Alpha (), or the level of significance. Alpha is some times referred to as the
amount of risk taken in an experiment. Alpha represents the proportion of the
area of the curve occupied by the rejection region. The most commonly used
values of  are 0.001, 0.01, 0.05 and 0.10. The larger the area of the rejection
region, the greater is the risk of committing Type I error.
Type II Error
A Type II error is committed by failing to reject a false null hypothesis. That is to
say that, accepting a null hypothesis when it is false is called a Type II error. The
probability of committing a Type II error is represented by beta ().
Alpha () is determined before the experiment, however, Beta () is computed
using alpha, the hypothesized parameter, and various theoretical alternatives to
the null hypothesis.
(Null Hypothesis)
State of Nature
Decision Ho True Ho False
Accept Ho Correct Decision Type II Error
Reject Ho Type I Error Correct Decision
There is a tradeoff between alpha and beta (Type I and Type II errors). The
probability of making one type of error can be reduced only if we are willing to
increase the probability of making the other type of error. However, this does
not mean that 1; rather it means that the smaller  is the larger will be ,
and the larger  in the smaller  will be.

4. 4
One – Tailed Vs Two – Tailed Tests
These are three possible null hypotheses along with their corresponding
alternative hypotheses.
(1) Ho:    (2) Ho:    (3) Ho:   
Ha:    Ha:   Ha:   
  
Leads to two – tailed test leads to a right – tailed test Leads to a left tests
Two – Tailed test
A two – tailed test of a hypothesis will reject the null hypothesis if the sample
statistic is significantly higher than or lower than a hypothesized population
parameter. Thus, in a two – tailed test there are two rejection regions. A two
tailed is appropriate when the null hypothesis is equal to some specified value
(e.g. Ho:   ) and the alternative hypothesis is different from (not equal to)
some specified value (e.g. Ha:  . The dividing point between the region
where the null hypothesis is rejected and accepted is called critical value.
One Tailed test
A one tailed test is one in which the alternative hypothesis is directional; unlike a
two – tailed test which does not specify direction.
One – tailed test can be:
 A right – tailed / upper – tailed test.
 A left – tailed / lower – tailed test.
Aright – tailed test will reject the null hypothesis if the sample statistic is
significantly higher than the hypothesized population parameter.
A left – tailed test will reject the null hypothesis if the sample statistic is
significantly lower than the hypothesized population parameter.
STEPS IN HYPOTHESIS TESTING
A summary of the steps that can be applied to any hypothesis test are:
1. Determine the null and alternative hypotheses.
E.g. Ho:   
Ha:   

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2. Select the test statistic that will be used to decide whether or not to reject the
null hypothesis
E.g. Z – distribution, t – distribution, F- dist, x2
– distribution
3. Select the level of significance to determine the critical values and develop the
rejection rule that indicates the values of the test statistic that will lead to the
rejection of Ho.
E.g.  = 0.05 Z025 = 1.96  Reject Ho if /Sample Z/  1.96
4. Collect the sample data, and compute the value the test statistic. A test
statistic is a random variable whose value is used to determine whether we
reject the null hypothesis.
E.g. Sample Z=2.0
5. Compare the value of the test statistic to the critical value(s) and make the
decision (either reject Ho or accept HO /do not reject).
HYPOTHESIS TEST ABOUT A POPULATION MEAN: Population - Normal,
Standard Deviation - Known
In hypothesis testing if the population in normal and standard deviation is
known, we use Z-Value to test the hypothesis; regardless of the sample size, n. It
is also applicable when n  3 regardless of the pop distribution
Example:
1. Matador-Addis Tyre Share Company claims that its tires have a mean life of
35,000 miles. A random sample of 16 of these tires is tested if the sample mean in
33,000 miles. Assume that the population standard deviation is 3000 miles and
the lives of tires are approximately normally distributed. Test the share
company’s claim using a 5% level of significance.
Solution
1. Ho:  = 35,000 miles 2. Z – distribution, two tailed test
Ha:  ≠ 35,000 miles
3.  = 0.05 4. X = 33,000 miles
/2 = 0.025  = 3,000 miles
Z0.025 = ± 1.96 n = 16 tires
Reject Ho if /Sample Z/ > 1.96 Sample Z =?

6. 6
67
.
2
16
000
,
3
000
,
35
000
,
33
000
,
33 



Z
5. Reject Ho because /-2.67/ > 1.96
2. A Teachers’ union is on strike for higher wages. The union claims that the mean
salary for teachers is at most Birr 8,400 per year. The legislator does not want to
reject the union’s claim, however, unless the evidence is very strong against if.
Assume that salaries follow a normal distribution and the population standard
deviation is known to be Birr 3000. A random sample of 64 teachers is obtained,
and the sample mean is Birr, 9,400. Test if the state legislator accepts the unions’
claim or not at 1% significance level.
Solution:
1. Ho:  ≤ Birr 8,400 2. Z – distribution, Right – tailed test
Ha:   Birr 8,400
3.  = 0.01
Z = Z0.01 = 2.33
Reject Ho if sample Z  + 2.33
4. n = 64
X = Birr 9,400
 = Birr 3,000
Sample Z =?
67
.
2
64
000
,
3
400
,
8
400
,
9
400
,
9 



Z
5. Reject Ho because + 2.67  2.33
3. A fertilizer company claims that the use of its product will result in a yield of at
least 35 quintals of wheat per hectare, on average, Application at the fertilizer to
a randomly selected 36 sample hectares resulted in a yield of 34quintals per
hectare. Assume the population standard deviation is 5 quintals and yields per
hectare are normally distributed. Test the company’s claim at 1% level of
significance.
Solution
1. Ho:  ≥ 35 quintals
Ha:   35 quintals
2. Z – distribution, left – tailed test
3.  = 0.01
Z= Z0.01 = 2.33
 Reject Ho if sample Z  -2.33

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4. X = 34 quintals
n = 36
 = 5 quintals
Sample Z =?
20
.
1
36
5
35
34
34 



Z
5. Do not reject H0,because -1.20 > -2.33
4. A survey of college graduates showed that the average yearly cash income for
these graduates in at least Birr 12,000. In Addis where you live this average does
not seem possible, so you decide to test this claim. You randomly select 48
graduates who are marking. The sample average income for these working
graduates is Birr 11,400 with a standard deviation of Birr 2,280. Is there enough
evidence from this sample data to reject the national claim for your area as being
too high? Use  = 0.10.
Solution
1. Ho :   Birr 12,000
Ha :   Birr 12,000
2. Z – distribution, Left tailed test
3.  = 0.1
Z = Z0.1 = -1.28
Reject Ho if sample Z < -1.28
4. X = Birr 11,400
S = Birr 2,280
n = 48
Sample Z =?
82
.
1
48
280
,
2
000
,
12
400
,
11
400
,
11 



Z
= -1.82
5. Reject Ho because –1.82 < -1.28