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Hypothesis Testing. Hypothesis Testing. Hypothesis is a claim or statement about a property of a population. Hypothesis Testing is to test the claim or statement Example : A conjecture is made that “the average starting salary for computer science gradate is Rs 45,000 per month”.

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slide2

Hypothesis Testing

  • Hypothesisis a claim or statement about a property of a population.
  • Hypothesis Testing is to test the claim or statement
  • Example: A conjecture is made that “the average starting salary for computer science gradate is Rs 45,000 per month”.
hypothesis testing1
Hypothesis Testing
  • Null Hypothesis (H0): is the statement being tested in a test of hypothesis.
  • Alternative Hypothesis (H1): is what is believe to be true if the null hypothesis is false.
null hypothesis
Null Hypothesis
  • Must contain condition of equality

=, ³, or £

  • Test the Null Hypothesis directly
  • Reject H0 or fail to reject H0
alternative hypothesis
Alternative Hypothesis
  • Must be true if H0 is false
  • ¹, <, >
  • ‘opposite’ of Null Hypothesis

Example

H0 : µ = 30 versus H1 : µ > 30 or

H1 : µ < 30

slide6

Identify the Problem

  • State the Null Hypothesis (H0: m³ 3)
  • State its opposite, the Alternative Hypothesis (H1: m < 3)
  • Hypotheses are mutually exclusive
slide7

Hypothesis Testing Process

Assumethe

population

meanage is 50.

(NullHypothesis)

Population

The Sample

Mean Is 20

No, not likely!

REJECT

Null Hypothesis

Sample

slide8

Reason for RejectingH0

Sampling Distribution

It is unlikely that we would get a sample mean of this value ...

... Therefore, we reject the null hypothesis that m = 50.

... if in fact this were the population mean.

m

= 50

Sample Mean

20

H0

slide9

Level of Significance,a

  • Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True
    • Called Rejection Region of Sampling

Distribution

  • Designateda(alpha)
    • Typical values are 0.01, 0.05, 0.10
  • Selectedby the Researcherat the Start
  • Provides theCritical Value(s)of the Test
slide10

Level of Significanceand the Rejection Region

Critical Value(s)

a

H0: m³ 3 H1: m < 3

Rejection Regions

0

a

H0: m£ 3 H1: m > 3

0

a/2

H0: m= 3 H1: m¹ 3

0

slide11

Errors in Making Decisions

  • Type I Error
    • Reject True Null Hypothesis
    • Has Serious Consequences
    • Probability of Type I Error is a
      • Called Level of Significance
  • Type II Error
    • Do Not Reject False Null Hypothesis
    • Probability of Type II Error Is b (Beta)
slide12

Result Possibilities

H0: Innocent

Hypothesis

Test

Jury Trial

Actual Situation

Actual Situation

Innocent

Guilty

H

True

H

False

Verdict

Decision

0

0

Do Not

Type II

a

Correct

Error

Innocent

1 -

Reject

b

)

Error (

H

0

Type I

Reject

Error

Correct

Guilty

Error

H

(1 -

)

b

0

)

(

a

type i error
Type I Error
  • The mistake of rejecting the null hypothesis when it is true.
  • The probabilityof doing this is called the significance level, denoted by a(alpha).
  • Common choices for a: 0.05 and 0.01
  • Example: rejecting a perfectly good parachute and refusing to jump
type ii error
Type II Error
  • The mistake of failing to reject the null hypothesis when it is false.
  • Denoted by ß (beta)
  • Example: Failing to reject a defectiveparachute and jumping out of a plane with it.
critical region
CriticalRegion
  • Set of all values of the test statistic that would cause a rejection of the null hypothesis.

Critical

Region

critical region1
Critical Region
  • Set of all values of the test statistic that would cause a rejection of the null hypothesis.

Critical

Region

critical region2
Critical Region
  • Set of all values of the test statistic that would cause a rejection of the null hypothesis.

Critical

Regions

critical value
Critical Value

Value (s) that separates the critical region from the values that would not lead to a rejection of H0.

Reject H0

Fail to reject H0

Critical Value

( z score )

conclusions in hypothesis tests
Conclusions in Hypothesis Tests

Original

claim is H0

(This is the

only case in

which the

original claim

is rejected).

“There is sufficient

evidence to warrant

rejection of the claim

that. . . (original claim).”

Yes

(Reject H0)

Do

you reject

H0?.

No

(Fail to

reject H0)

“There is not sufficient

evidence to warrant

rejection of the claim

that. . . (original claim).”

Original

claim is H1

(This is the

only case in

which the

original claim

is supported).

Yes

(Reject H0)

“The sample data

supports the claim that

. . . (original claim).”

Do

you reject

H0?

No

(Fail to

reject H0)

“There is not sufficient

evidence to support

the claim that. . .

(original claim).”

left tailed test
Left-tailed Test

H0: µ ³ 200

H1: µ < 200

Points Left

Reject H0

Fail to reject H0

Values that

differ significantly

from 200

200

right tailed test
Right-tailed Test

H0: µ £ 200

H1: µ > 200

Points Right

Reject H0

Fail to reject H0

Values that

differ significantly

from 200

200

two tailed test
Two-tailed Test

a is divided equally between

the two tails of the critical

region

H0: µ = 200

H1: µ ¹ 200

Means less than or greater than

Reject H0

Reject H0

Fail to reject H0

200

Values that differ significantly from 200

definition
Definition

Test Statistic:

is a sample statistic or value based on sample data

Example:

x–µx

z=

s /

n

slide24

Question:

How can we justify/test this conjecture?

A. What do we need to know to justify this conjecture?

B. Based on what we know, how should we justify this conjecture?

slide25

Answer to A:

Randomly select, say 100, computer science graduates and find out their annual salaries

---- We need to have some sample observations, i.e., a sample set!

slide26

Answer to B:

That is what we will learn in this chapter

---- Make conclusions based on the sample observations

slide27

Statistical Reasoning

Analyze the sample set in an attempt to distinguish between results that can easily occur and results that are highly unlikely.

figure 7 1 central limit theorem1

Likely sample means

µx = 30k

Figure 7-1 Central Limit Theorem:

Distribution of Sample Means

Assume the conjecture is true!

figure 7 1 central limit theorem2

Likely sample means

µx = 30k

z= –1.96

x = 20.2k

z = 1.96

x= 39.8k

or

or

Figure 7-1 Central Limit Theorem:

Distribution of Sample Means

Assume the conjecture is true!

figure 7 1 central limit theorem3

Likely sample means

µx = 30k

z= –1.96

x = 20.2k

z = 1.96

x= 39.8k

or

or

Figure 7-1 Central Limit Theorem:

Distribution of Sample Means

Assume the conjecture is true!

Sample data: z= 2.62

x= 43.1k

or

problem 1
Problem 1

Test µ = 0 against µ > 0, assuming normally and using the sample [multiples of 0.01 radians in some revolution of a satellite]

1, -1, 1, 3, -8, 6, 0 (deviations of azimuth)

Choose α = 5%.

problem 2
Problem 2

In one of his classical experiments Buffon obtained 2048 heads in tossing a coin 4000 times. Was the coin fair?

problem 3
Problem 3

In one of his classical experiments K Pearson obtained 6019 heads in 12000 trials. Was the coin fair?

problem 5
Problem 5

Assuming normality and known variance б2 = 4, test the hypotheses µ = 30 using a sample of size 4 with mean X = 28.5 and choosing α = 5%.

problem 7
Problem 7

Assuming normality and known variance б2 = 4, test the hypotheses µ = 30 using a sample of size 10 with mean X = 28.5. What is the rejection region in case of a two sided test with α= 5%.

problem 9
Problem 9

A firm sells oil in cans containing 1000 g oil per can and is interested to know whether the mean weight differs significantly from 1000 g at the 5% level, in which case the filling machine has to be adjusted. Set up a hypotheses and an alternative and perform the test, assuming normality and using a sample of 20 fillings with mean 996 g and Standard Deviation 5g.

problem 11
Problem 11

If simultaneous measurements of electric voltage by two different types of voltmeter yield the differences (in volts)

0.8, 0.2, -0.3, 0.1, 0.0, 0.5, 0.7 and 0.2

Can we assert at the 5% level that there is no significant difference in the calibration of the two types of instruments? Assume normality.