Testing Hypotheses Pri

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Testing Hypotheses


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Presentation Outline Hypothesis

Types of Hypotheses

Null and Alternative Hypothesis

Motivation for Hypothesis Testing

Two-tailed and One-tailed Hypothesis

Test Statistic

Decision Errors

Decision Rules

Two-tailed and One-tailed Tests

Test of hypothesis for a population mean (Z Test , t Test)

Hypothesis Test for a Proportion


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What is Hypothesis ??

A hypothesis is a statement that something is true.A hypothesis is an assumption about the populationparameter.

A parameter is a characteristic of the population, likeits mean or variance.

The parameter must be identified before analysis.

I assume the mean height ofstudents in this class is 5 feet 4

inches.


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Hypothesis Testing

Statisticians follow a formal process to determinewhether to reject a null hypothesis, based on sample

data.

This process is called hypothesis testing.


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Types of Hypotheses


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Null Hypothesis

A hypothesis to be tested.

The symbol H0is used to represent the null hypothesis

Alternative Hypothesis

A hypothesis which is alternativ

e to the null hypothesis.

The symbol HAor H1is used to represent the alternativehypothesis.

It is the hypothesis that is believed to be true, or what

you are trying to prove is true.


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Null Hypothesis Vs.Alternative HypothesisNull Hypothesis Alternative Hypothesis

Begin with the assumption that thenull hypothesis is TRUE

Is generally the hypothesis that isbelieved to be true by the

researcher

Always contains the = sign Never contains the = sign

Null Hypothesis may or may not berejected

Alternative Hypothesis may or maynot be accepted

Observed difference is due tochance and there is nosignificant

difference.

Observed difference is real


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Example: Null Hypothesis and Alternative Hypothesis

NumberedTickets

The average ofthe box equals 50

The average of thebox is less than 50

Mr. Null Mr. Alt

Null Hypothesis Alternative Hypothesis

Arguing about the averageof a large box of numberedtickets


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Motivation for Hypothesis Testing

The intent of hypothesis testing is formally examine twoopposing hypotheses, H0 and HA.

These two hypotheses are mutually exclusive so that oneis true to the exclusion of the other.

We accumulate evidence - collect and analyze sample

information - for the purpose of determining which ofthe two hypotheses is true and which of the twohypotheses is false.


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One-tailed Hypothesis

A Hypothesis in which the value of a parameter is specifiedas being either: above a certain value, or below a certain value.Eg: Mean height of students is less than 5 feet 4inches

Two-tailed Hypothesis

A form of alternative hypothesis in which a populationparameter is different than a specified value.

Example: Mean height of students is not 5 feet 4inches

0: aH

0:

aH


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Test Statistic

i. The decision to reject or fail to reject is based oninformation contained in a sample drawn from thepopulation of interest.

ii. The sample values are used to compute a single number,

which operates as a decision maker.

iii. This decision maker is called test statistic

A test statistic is used to measure the differencebetween the data and what is expected on the null

hypothesis.


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Determining Rejection and Acceptance Region

Rejection Region

If test

statistic fallsin someinterval whichsupport

alternativehypothesis, wereject the nullhypothesis.

AcceptanceRegion

It test

statistic fallsin someinterval whichsupport null

hypothesis, wefail to rejectthe nullhypothesis.

Critical Value

The value of

the point,which dividethe rejectionregion and

acceptance one


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Decision Errors

Type I Error

Reject Null Hypothesis when it is true

Probability of Type I Error is denoted by alphaa

The probability of committing a Type I error is

called thesignificance level.


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Decision Errors

Type II Error

Researcher fails to reject a null hypothesisthat is false

Probability of Type II Error is denoted byBeta,

The probability of not committing a Type II

error is called the Power of the test.


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Decision Rules

In practice, statisticians describe decision rules for

rejecting the null hypothesis in two ways with reference to a P-value or

with reference to a region of acceptance.

P-value The strength of evidence in support of a null hypothesis is measured

by the P-value.

Suppose the test statistic is equal to S. The P-value is theprobability of observing a test statistic as extreme as S, assumingthe null hypothesis is true.

If the P-value is less than the significance level, we reject the null

hypothesis.


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Decision Rules

Rejection region If test statistic falls in some interval which support

alternative hypothesis, we reject the null hypothesis.

The set of values outside the region of acceptance.

Acceptance Region The region of acceptance is a range of values. If the test

statistic falls within the region of acceptance, the nullhypothesis is not rejected.

It test statistic falls in some interval which support nullhypothesis, we fail to reject the null hypothesis.

The value of the point, which divide the rejection regionand acceptance one is called

Critical value


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Two-tailed vs. One-tailed TestsTwo-tailed Test One-tailed Test

Test of deciding whether a

population parameter is different

thana specified value

Test is about whether a population

parameter is less than a specified

value (left-tailed test) or more

than a specified value (right-

tailed test)In a two-tailed test, the direction

of the difference is not predicted.

In a one-tailed test, the researcher

predicts the direction (i.e.

greater or less than) of the

difference.

A two-tailed test splits the critical

region equally on both sides of

the curve.

All of the critical region is placed

on one side of the curve in the

direction of the prediction.


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1. Test of hypothesis for a population mean

Two-Tailed Test

Large Sample

With reference to region of acceptance orrejection


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Example - The population of all minority workers has a

mean wage of $14,500 with a standard deviation of $200.

Test whether a sample of 100 having an average of $14,300

and = .05 differs from the population average.

Step 1 Stating the null and alternative hypothesis

1. HO : $14,500

HA : $14,500


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Step 2 - Select Sampling Distribution and Establish theCritical Region

Critical Region begins at Z= 1.96

This is the critical Z score associated with =

.05, two-tailed test.

If the obtained Z score falls in the Critical

Region, or the region of rejection, then we

would reject the H0.


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Step 3:

Applying the Formula to Compute the Test Statistic (Z

for large samples ( 100)

N

Z

When the Population Standard deviation is not knownthe formula is:

1

Ns

Z

StandardError

Sample SD


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4.zx

SE

14,300 14,500

200

100

10

Substituting the values into the formula, we calculate aZ score of -10

Rule of Thumb:

If the test statistic is in the Critical Region ( =.05, beyond1.96):Rejectthe H0. The difference is significant.

If the test statistic is not in the Critical Region (at =.05,between +1.96 and -1.96):Fail to rejectthe H0. The difference is not significant.


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Step 5

Make a Decision and Interpret Results

The obtained Z (-10) score falls in the Critical Region,

so we rejectthe H0. Therefore, the H0 is false and must be rejected. There

is evidence that the salaries are different.

C C

Z = -10Rejection region


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2. Test of hypothesis for a population mean

Two-Tailed Test

Small Sample

With reference to region of acceptance orrejection


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Example - A manufacturer of ball bearings have a diameter of

.25 inches and a sample standard deviation of .05 inches. A

random sample ofn = 25 ball bearings reveal a mean diameter

of .2670 inches. Conduct a hypothesis test at the 10 % level ofsignificance to determine whether there is statistical

significance that the manufacturing process is not running

correctly, that is .25 inches.

Step 1 Stating the null and alternative hypothesis

1. HO

: .25

HA

: .25


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Step 2

n< 30

unknown

The appropriate t distribution has (n- 1) Degrees offreedomDegrees of Freedom = n- 1 = 25 - 1 = 24

Step 3 Determining the Critical value

Given significance level of 0.10 and degree of freedom of 24,we look in the t distribution table.

Under 0.10 column until we reach 24 degrees of freedom row.

There we find the critical value of t, which is 1.711

Thus we use t distribution.


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Step 4 Calculating the t - statistic

4. tx

SE

.2670 .25

.05

25

1.7

Critical valuet = - 1.711

Critical valuet = 1.711

t = 1.7Acceptanceregion

Step 5 Sketching the distribution and marking the

sample value and critical value Step 6 Interpreting theResult

The sample mean falls in

the acceptance region,hence Fail to reject HO

There is not enoughevidence that it is not

running incorrectly.


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Main Considerations in Hypothesis Testing

Sample size

Use Z for large samples, t for small (


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The Curve for Two- vs. One-tailed Tests at = .05

Two-tailed test:is there a significant difference?

One-tailed tests:

is the sample meangreater than ?

is the sample meanless than ?


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Type I and Type II Errors

Type I, or alpha error:

Rejecting a true null hypothesis

Type II, or beta error:

Failing to reject a false null hypothesis


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START

Decide whether this is a two-tailed or a one-tailed test. State your hypothesis. Select a

level of significance appropriate for this decision

Decide which distribution(tor z)is appropriate and find the critical value for the chosen

level of significance from the appropriate appendix table.

Circulate the standard error of the sample statistic. Use the standard error to

standardize the observed sample value.

Sketch the distribution and mark the position of the standardized sample value and the

critical values for the test

Is the sample

statistic within

the acceptance

region?AcceptHo

Translate the statistical results

into appropriate managerial action

Accept

Ho

Yes No


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Hypothesis Test for a Proportion

Two-Tailed Test

With reference to P-Value

The CEO of a large electric utility claims that 80 % of his1,000,000 customers are very satisfied with the service

they receive. To test this claim, the local newspaper

surveyed 100 customers, using simple random sampling.

Among the sampled customers, 73 % say they are verysatisified. Based on these findings, can we reject the CEO's

hypothesis that 80% of the customers are very satisfied?

Using a 0.05 level of significance.


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Solution

Step 1

Stating the null hypothesis and an alternative hypothesis.

Null hypothesis : P = 0.80

Alternative hypothesis : P 0.80


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Step 2

Deciding the following:

a. Nature of the testTwo-tailed test

b. Level of Significance 0.05

c. Choosing the distribution Normal distribution(as n > 30) Z-test


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Step 3 - Analyse sample data

Using sample data, we calculate the standard deviation () and

compute the z-score test statistic (z).

= square root[ P * ( 1 - P ) / n ]

= sqrt [(0.8 * 0.2) / 100] = sqrt(0.0016) = 0.04

z = (p - P) /

= (.73 - .80)/0.04 = -1.75

Where, P is the hypothesized value of population proportion in the null

hypothesis p is the sample proportion n is the sample size.


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Step 4 Determimg P Value

P-value is the probability that the z-score is less than -1.75 or greater than 1.75.

We use the Normal Distribution Calculator to find

P(z < -1.75) = 0.04

P(z > 1.75) = 0.04Thus, the P-value = 0.04 + 0.04 = 0.08


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Step 5 - Interpreting the results

Ifpvalue > a, Do Not Reject H0

Ifpvalue < a, Reject H0

Since the P-value (0.08) is greater than the significance

level (0.05), we cannot reject the null hypothesis.

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Testing Hypotheses Pri