Hypothesis Testing

In this session ….

 What is hypothesis testing?
 Interpreting and selecting significance level
 Type I and Type II errors
PROBABILITY DISTRIBUTIONS
 One tailed and two tailed tests
 Hypothesis tests for population mean
 Hypothesis tests for population proportion
 Hypothesis tests for population standard deviation

What is Hypothesis Testing?

Hypothesis testing refers to
1. Making an assumption, called hypothesis, about a population
parameter. - the B-school
2. Collecting sample data.
3. Calculating a sample statistic.
4. Using the sample statistic to evaluate the hypothesis (how likely
is it that our hypothesized parameter is correct. To test the
validity of our assumption we determine the difference between
the hypothesized parameter value and the sample value.)

HYPOTHESIS
TESTING

Null hypothesis, H0 Alternative hypothesis,HA
State the hypothesized value of the All possible alternatives other than
parameter before sampling. the null hypothesis.
The assumption we wish to test E.g µ ≠ 20
(or the assumption we are trying to
reject) µ > 20
E.g population mean µ = 20 µ < 20
 There is no difference between There is a difference between coke
coke and diet coke and diet coke

Null Hypothesis

The null hypothesis H0 represents a theory that has been put
forward either because it is believed to be true or because it
- the B-school
is used as a basis for an argument and has not been proven.
For example, in a clinical trial of a new drug, the null
hypothesis might be that the new drug is no better, on
average, than the current drug. We would write
H0: there is no difference between the two drugs on an
average.

Alternative Hypothesis

The alternative hypothesis, HA, is a statement of what a statistical
hypothesis test is set up to establish. For example, in the clinical
- the B-school
trial of a new drug, the alternative hypothesis might be that the
new drug has a different effect, on average, compared to that of the
current drug. We would write
HA: the two drugs have different effects, on average.
or
HA: the new drug is better than the current drug, on average.

The result of a hypothesis test:
‘Reject H0 in favour of HA’ OR ‘Do not reject H0’

Selecting and interpreting significance level

1. Deciding on a criterion for accepting or rejecting the null
hypothesis.
- the B-school
2. Significance level refers to the percentage of sample means that
is outside certain prescribed limits. E.g testing a hypothesis at
5% level of significance means
 that we reject the null hypothesis if it falls in the two regions of
area 0.025.
 Do not reject the null hypothesis if it falls within the region of
area 0.95.
3. The higher the level of significance, the higher is the probability
of rejecting the null hypothesis when it is true.
(acceptance region narrows)

Type I and Type II Errors

1. Type I error refers to the situation when we reject the null hypothesis
when it is true (H0 is wrongly rejected).
- the B-school
e.g H0: there is no difference between the two drugs on average.
Type I error will occur if we conclude that the two drugs produce
different effects when actually there isn’t a difference.
Prob(Type I error) = significance level = α
2. Type II error refers to the situation when we accept the null
hypothesis when it is false.
H0: there is no difference between the two drugs on average.
Type II error will occur if we conclude that the two drugs produce
the same effect when actually there is a difference.
Prob(Type II error) = ß

Type I and Type II Errors – Example

Your null hypothesis is that the battery for a heart pacemaker
has an average life of 300 days, with the alternative
- the B-school
hypothesis that the average life is more than 300 days. You are
the quality control manager for the battery manufacturer.
(a)Would you rather make a Type I error or a Type II error?
(b)Based on your answer to part (a), should you use a high or
low significance level?

Type I and Type II Errors – Example

Given H0 : average life of pacemaker = 300 days, and HA:
Average life of pacemaker > 300 days
- the B-school
(a)It is better to make a Type II error (where H0 is false i.e
average life is actually more than 300 days but we accept H0
and assume that the average life is equal to 300 days)
(b)As we increase the significance level (α) we increase the
chances of making a type I error. Since here it is better to
make a type II error we shall choose a low α.

Two Tail Test

Two tailed test will reject the null hypothesis if the sample mean
is significantly higher or lower than the hypothesized mean.
- the B-school
Appropriate when H0 : µ = µ0 and HA: µ ≠ µ0
e.g The manufacturer of light bulbs wants to produce light bulbs
with a mean life of 1000 hours. If the lifetime is shorter he will lose
customers to the competition and if it is longer then he will incur
a high cost of production. He does not want to deviate
significantly from 1000 hours in either direction. Thus he selects
the hypotheses as
H0 : µ = 1000 hours and HA: µ ≠ 1000 hours
and uses a two tail test.

One Tail Test

A one-sided test is a statistical hypothesis test in which the values
for which we can reject the null hypothesis, H0 are located entirely
in one tail of the probability distribution.
- the B-school
Lower tailed test will reject the null hypothesis if the sample mean
is significantly lower than the hypothesized mean. Appropriate
when H0 : µ = µ0 and HA: µ < µ0
e.g A wholesaler buys light bulbs from the manufacturer in large
lots and decides not to accept a lot unless the mean life is at least
1000 hours.
H0 : µ = 1000 hours and HA: µ <1000 hours
and uses a lower tail test.
i.e he rejects H0 only if the mean life of sampled bulbs is
significantly below 1000 hours. (he accepts HA and rejects the lot)

One Tail Test

Upper tailed test will reject the null hypothesis if the sample mean
is significantly higher than the hypothesized mean. Appropriate
- the B-school
when H0 : µ = µ0 and HA: µ > µ0
e.g A highway safety engineer decides to test the load bearing
capacity of a 20 year old bridge. The minimum load-bearing
capacity of the bridge must be at least 10 tons.
H0 : µ = 10 tons and HA: µ >10 tons
and uses an upper tail test.
i.e he rejects H0 only if the mean load bearing capacity of the
bridge is significantly higher than 10 tons.

Hypothesis test for population mean

n ( x  0 )
H0 : µ = µ0 and Test statistic  
s
- the B-school
For HA: µ > µ0, reject H0 if   t n 1,

For HA: µ < µ0, reject H0 if   t n 1,

For HA: µ ≠ µ0, reject H0 if   tn1, 2

For n ≥ 30, replace t n 1, by z


A weight reducing program that includes a strict diet and exercise
claims on its online advertisement that it can help an average
- the B-school
overweight person lose 10 pounds in three months. Following the
program’s method a group of twelve overweight persons have lost
8.1 5.7 11.6 12.9 3.8 5.9 7.8 9.1 7.0 8.2 9.3 and 8.0 pounds
in three months. Test at 5% level of significance whether the
program’s advertisement is overstating the reality.


Solution:
H0: µ = 10 (µ0) HA: µ < 10 (µ0)
n = 12, x(bar) = 8.027, s = 2.536,  = 0.05
12(8.075  10) 3.46  1.925
   2.62
2.536 2.536
Critical t-value = -t n-1,α= - t11,0.05= -2. 201 (TINV)

Since  < -t n-1,α we reject H0 and conclude that the program
is overstating the reality.
(What happens if we take  = 0.01? Is the program
overstating the reality at 1% significance level?)

Hypothesis test for population proportion

n ( p  p0 )
ˆ
H0 : p = p0 and Test statistic  
p0 (1  p0 )
- the B-school
For HA: p > p0 reject H0 if   z

For HA: p < p0 reject H0 if   z

For HA: p ≠ p0 reject H0 if   z 2


A ketchup manufacturer is in the process of deciding whether to
- the B-school
produce an extra spicy brand. The company’s marketing research
department used a national telephone survey of 6000 households
and found the extra spicy ketchup would be purchased by 335 of
them. A much more extensive study made two years ago showed
that 5% of the households would purchase the brand then. At a 2%
significance level, should the company conclude that there is an
increased interest in the extra-spicy flavor?


335
n  6000, p
ˆ  0.05583
6000
- the B-school
H0 : p  0.05( p0 ) H A : p  0.05
n ( p  p0 )
ˆ 6000  0.00583
 
p0 (1  p0 ) 0.05  0.95
77.459  0.00583
  2.072
0.218
  0.02
Z (the critical value of Z )  2.05 (NORMSINV)
   Z we reject H0 i.e the current interest is significantly greater
than the interest of two years ago.

Hypothesis test for population standard
deviation
(n  1)s 2
H0 :  =  0 and Test statistic  
 02
- the B-school
For HA:  > 0 reject H0 if   (2(1),
n
R)

For HA:  < 0 reject H0 if   (2(1),1
n
R)

For HA:  ≠ 0 reject H0 if   (2(1),1 2 or   ( n1), 2
R) 2( R )
n

Hypothesis test for comparing two population
means
Consider two populations with means µ1, µ2 and standard deviations 1 and 2.
x  1 and x  2 are the means of the sampling distributions of population1
1 2

and population2 respectively.  x and  x denote the standard errors of the
- the B-school
1

sampling distributions of the means.
2

 12  2
2
x1  x2 is the mean of the difference between sample means and X  X  
1 2
n1 n2
is the corresponding standard error.
( X  X 2 )  ( 1  2 )H
H0 : µ1 = µ2 and test statistic,   1 0

 x x
For HA: µ1 > µ2 reject H0 if  > Z
1 2

Here  denotes the standardized
For HA: µ1 < µ2 reject H0 if  <- Z difference of sample means
For HA: µ1  µ2 reject H0 if   Z 2
(decision makers may be concerned with parameters of two populations e.g do
female employees receive lower salary than their male counterparts for the
same job)

Hypothesis test for comparing population means

A sample of 32 money market mutual funds was chosen on
- the B-school
January 1, 1996 and the average annual rate of return over the
past 30 days was found to be 3.23% and the sample standard
deviation was 0.51%. A year earlier a sample of 38 money-market
funds showed an average rate of return of 4.36% and the sample
standard deviation was 0.84%. Is it reasonable to conclude (at α
= 0.05) that money-market interest rates declined during 1995?

Hypothesis test for comparing population means

n1  32, x1  3.23,  1  0.51 n2  38, x2  4.36,  2  0.84
H0 : 1  2 H A : 1  2

 X X 
 12

 22
 
- the B-school
0.26 0.71
 0.026  0.163
1 2
n1 n2 32 38
( x1  x2 )  ( 1  2 )H0 1.13  0
   6.92
 X X1 2
0.163
  0.05
Critical value of Z  Z  1.64
   Z we reject H0 and conclude that there has
been a decline.

Hypothesis test for comparing population
proportions
Consider two samples of sizes n1 and n2 with p1 and p2 as the respective
proportions of successes. Then
n1p1  n2 p2
p
ˆ
n1  n2
- the B-school
is the estimated overall proportion of successes in the two
populations.
ˆˆ ˆˆ
pq pq is the estimated standard error of the difference
 p p 
ˆ 
1 2
n1 n2 between the two proportions.
( p1  p2 )  ( p1  p2 )H0
H0 : p1 = p2 and test statistic,  
 x x
ˆ 1 2
For HA: p1 > p2 reject H0 if  > Z
A training director may wish to
For HA: p1 < p2 reject H0 if  <- Z determine if the proportion of
For HA: p1  p2 reject H0 if   Z 2 promotable employees at one office is
different from that of another.

proportions

A large hotel chain is trying to decide whether to convert more of
its rooms into non-smoking rooms. In a random sample of 400
- the B-school
guests last year, 166 had requested non-smoking rooms. This year
205 guests in a sample of 380 preferred the non-smoking rooms.
Would you recommend that the hotel chain convert more rooms to
non-smoking? Support your recommendation by testing the
appropriate hypotheses at a 0.01 level of significance.

proportions
166 205
n1  400, p1   0.415, n2  380, p2   0.5395
400 380
H0 : p1  p2 H A : p1  p2

p
ˆ 
- the B-school
n1p1  n2 p2 400  0.415  380  0.5395 (Proportion of success
 0.4757 in the two populations)
n1  n2 400  380
1 1  1 1 
 p  p  pq 
ˆ ˆˆ    0.4757  0.5243     0.0358
1 2
 n1 n2   400 380 
  0.01 The hotel chain should
Critical value of Z  Z  2.32 convert more rooms to non-
smoking rooms as there has
( p1  p2 )  ( p1  p2 )H0 0.1245  0
   3.48 been a significant increase in
 p1  p2
ˆˆ ˆ 0.0358 the number of guests
   Z we reject H seeking non-smoking rooms.
 0

Hypothesis Testing

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