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Concepts Description Hypothesis

Theory Laws Model

organize surprise

validateformalize

The Scientific Method

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Hypothesis Testing

• Population parameter = hypothesized?

• One sample mean = another sample mean?

• Null hypothesis

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Hypothesis Testing

• One-sample tests

– One-sample tests for the mean

– One-sample tests for proportions

• Two-sample tests

– Two-sample tests for the mean

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Hypothesis Testing

• Confidence interval

Interval

• Hypothesis testing

Particular, predetermined value

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Hypothesis Testing

• Hypothesis testing

Null hypothesis

• Purpose

Test the viability

• Null hypothesis

Population parameter

Reverse of what the experimenter believes

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Hypothesis Testing

1. State the null hypothesis, H0

2. State the alternative hypothesis, HA

3. Choose a, our significance level

4. Select a statistical test, and find the observed test

statistic

5. Find the critical value of the test statistic

6. Compare the observed test statistic with the critical

value, and decide to accept or reject H0

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Hypothesis Testing – Step 1

1. State the null hypothesis (H0)

– H0: μ = μ0

– H0: μ - μ0 = 0

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Hypothesis Testing – Step 2

2. State the alternative hypothesis

– HA: μ # μ0 two-sided (two-tailed)

or

– HA : μ > μ0

– HA : μ < μ0

one-sided (one-tailed)

upper-tailed

lower-tailed

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Hypothesis Testing – Step 3

3. Choose α, our significance level

– It really depends on what we are testing

– α = 0.05

– α = 0.01

– Type I error

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Hypothesis Testing - Errors

• Type I Error - α error, occurs when we reject

the null hypothesis when we should accept it

• Type II Error - β error, occurs when we

accept the null hypothesis when we should

reject it

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Hypothesis Testing - Errors

H0 is true H0 is false

Accept H0 Correct decision Type II Error (β)

(1-α)

Reject H0 Type I Error (α) Correct decision

(1-β)

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Hypothesis Testing – Step 4

4. Select a statistical test, and find the test statistic

Test statistic = - 0

Std. error

n

xz

/

ns

xz

/

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Hypothesis Testing – Step 4

4. Select a statistical test, and find the test statistic

Test statistic = - 0

Std. error

n

xt

/

ns

xt

/

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Hypothesis Testing – Step 5

5. Find the critical value of the test statistic

– Standard normal table

– Student’s t distribution table

– Two-sided vs. one-sided

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Two-sided tests Zα/2

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One-sided tests Zα

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Hypothesis Testing – Step 6

6. Compare the observed test statistic with the critical value

| Ztest | > | Zcrit | HA

| Ztest | | Zcrit | H0

Zcrit-Zcrit H0

HA HA

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| Ztest | > | 1.96 | HA

| Ztest | | 1.96 | H0

1.96-1.96 H0

HA HA

Hypothesis Testing – Step 6

6. Compare the observed test statistic with the critical value

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Hypothesis Testing – Step 6

Ztest > Zcrit HA

Ztest Zcrit H0

ZcritH0

HA

6. Compare the observed test statistic with the critical value

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Hypothesis Testing – Step 6

Ztest > 1.645 HA

Ztest 1.645 H0

1.645H0

HA

6. Compare the observed test statistic with the critical value

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p-value

• p-value is the probability of getting a value of the test

statistic as extreme as or more extreme than that observed

by chance alone, if the null hypothesis H0, is true.

• It is the probability of wrongly rejecting the null

hypothesis if it is in fact true

• It is equal to the significance level of the test for which

we would only just reject the null hypothesis

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p-value

• p-value vs. significance level

• Small p-values the null hypothesis is unlikely to be

true

• The smaller it is, the more convincing is the rejection of

the null hypothesis

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One-Sample z-Tests