Level of Significance, a and the Rejection Region
Critical
Value(s)
H0:
H1:
<3
H0:
H1:
H0:
3
>3
0
0
=3
H1:
11
Rejection
Regions
30-09-2023
/2
3
0
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Errors in Making Decisions
•
When using a sample statistic to make decisions about a
population parameter, there is a risk that you will reach an
incorrect conclusion.
368 gm
z
Weight of cereal
12
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Errors in Making Decisions
• Type I Error
• Rejected True Null Hypothesis which should not be rejected
• Has Serious Consequences
• Probability of Type I Error Is (risk level specified before)
o Called Level of Significance
o Confidence coefficient is
• Type II Error
• Not Rejected False Null Hypothesis
• Probability of Type II Error Is
• difference between hypothesized and the actual population value
• Power of a statistical test is
13
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Type I and Type II Error
Statistical decision
Actual decision
Ho is true
Ho is false
Not rejected Ho
Correct decision
Confidence =
Type II error
Probability =
Rejected Ho
Type I error
Probability =
Correct decision
Power =
If true population mean is 330gm. Then
is small that we conclude
mean has not changed from 368gm.
14
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Factors Affecting Type II Error, b
•
True Value of Population Parameter
― Increases when difference between Hypothesized
parameter & True value decreases
•
Significance Level ( )
•
15
Increases when
Decreases
•
Sample Standard Deviation (s)
― Increases when s Increases
•
Sample Size (n)
― Increases when n Decreases
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Hypothesis Testing: Steps
Assumption: The true mean no. of TVs in Indian homes is at least 3.
1. State H0
2. State H1
3. Choose
4. Choose n
5. Choose Test
6. Set Up Critical Value(s)
7. Collect Data
8. Compute Test Statistic
9. Make Statistical Decision
10. Express Decision
16
30-09-2023
H0 :
3
H1 :
3
= .05
n = 100
Z Test
Z = -1.645
100 households surveyed
Computed Test Statistic = -2
Reject Null Hypothesis
The true mean no. of TVs is less
than 3 in Indian households
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
•
•
•
•
•
•
•
•
17
Hypothesis Testing
A hypothesis is an assumption regarding a parameter
Hypothesis Testing is a formal statistical procedure to accept or reject the
hypothesis
The null hypothesis, H0 , is an assumption about the parameter.
The alternative hypothesis, H1, is the opposite of H0.
The testing procedure samples the population to test the two competing
statements H0 and H1.
Given the Null Hypothesis H0 and the Alternative Hypothesis H1
•
•
Data is collected to see whether H0 can be rejected
There are two possible conclusions
1.
There is enough evidence to reject H0 (and accept H1), or
2.
There is not enough evidence to reject H0
In which situation will a decision be made?
Usually that situation will suggest H1
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Hypothesis Testing: Developing Hypotheses
A new teaching method is developed that is claimed to be better
• H1: The new teaching method is better
• H0 : The new method is no better than the old method
A new drug is developed to lower blood sugar more than the existing drug
• H1 : The new drug lowers blood sugar more than the existing drug
• H0 : The new drug does not lower blood sugar more than the existing drug
The label on a coffee states that it contains 500g. For the Consumer Forum
• H0 : The label is correct. μ ≥ 500g
• H1 : The label is incorrect. μ < 500g
The label on a coffee can states that it contains 500g. For the Quality Inspector
• H0 : The label is correct. μ = 500g
• H1 : The label is incorrect. μ ≠ 500g
18
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Forms for Null and Alternative Hypotheses
•
•
•
•
•
•
•
19
The equality part of the hypotheses always appears in the null hypothesis.
H0 and Ha take one of the following three forms:
• H0: μ ≥ μ0 & H1: μ < μ0, One-Tailed Test (Lower Tail or Left Tail)
• H0: μ ≤ μ0 & H1: μ > μ0, One-Tailed Test (Upper Tail or Right Tail)
• H0: μ = μ0 & H1: μ ≠ μ0, Two-Tailed Test
• μ0 is the hypothesized value of the population mean
The CFO will shut down production if he feels μ > 12 liters
The volume dispensed is a random variable. So some bottles have less than 12
liters and others have more. Chances are that no bottle has exactly 12 liters of
water
Since hypothesis tests are based on sample data, there is the possibility of errors
Type I Error: The calibration is prefect. But the average of the sample of 36
bottles is much greater than 12 liters. Production is shut down But actually
nothing was wrong
Type II Error: The machine is in a bad shape and almost every bottle contains
much more than 12 liters.. But the sample average is less than 12 liters – since
the volume dispensed is random
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Three approaches for testing Hypothesis
• Critical Value Approach
If the test statistic falls into the nonrejection region, you do not reject
the null hypothesis.
o If the test statistic falls into the rejection region, you reject the null
hypothesis
o This is more intuitive
o
• p-value Approach
If the p-value is greater than or equal to α (p≥α), then do not reject the
null hypothesis.
o If the p-value is less α (p<α), then reject the null hypothesis
o
• Confidence Intervals
If the hypothesized value is contained within the interval, you do not
reject the null hypothesis
o If the hypothesized value does not fall into the interval, you reject the
null hypothesis
o
20
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
Hypothesis Testing: Example
Ozone sells 12 litre bottles of mineral water. Vice president (operations) believes
that excess water is being dispensed. If that is the case, he wants to shut down
production for a major overhaul of the machinery. 36 bottles were sampled, and
the mean volume of water was found to be 12.17. The population standard
deviation is believed to be 0.6 liters.
H0: μ ≤ 12
H1: μ > 12
Sample Mean = 12.17
How many σ’s is the observed mean from the hypothesised μ?
α=0.05
.
.
Question: Is this a too far or near enough?
If cut off is 1.645 then H0 is rejected
The cut off is usually implied by providing the significance level α
The cut off is then computed by referring the tables of the sampling distribution
21
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD
The Critical Value Approach I
1. The hypotheses: H0: μ ≤ 12 Vs H1: μ > 12
2. Data: n = 36, = 12.17,  = 0.6, α = 5%
3. Right-Tail Test (or Upper Tail Test)
4. Sampling Distribution:
5. Test Statistic:
.
. /
⁄
=
. /
~ N(0, 1)
= 1.7
6. Critical Value = 1.645 & Critical Region: z ≥
1.645
7. Since Test Statistic = 1.7 > 1.645 = Critical
Value, Test Statistic falls in critical region, we
reject H0.
α = .05
0
1.645
z
There is sufficient statistical evidence to infer
that the process is not meeting the target of 12
litres
22
30-09-2023
MMZG515 / QMZG515 Quantitative Methods
BITS Pilani, WILPD