Statistical Hypothesis

TestsNotes of STAT6205 by Dr. Fan

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Overview• Introduction of hypotheses tests (Sections 7.1,7.2)

o General logic

o Two types of error

o Parametric tests for one mean and for proportions

o What is the best test for a given situation?

• Order Statistics (Section 8.3)

• Wilcoxon tests (Section 8.5)

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Statistical Hypotheses• A statistical hypothesis is an assumption or

statement concerning one or more population parameters.

• Simple vs. composite hypotheses

E.g. A pharmaceutical company wants to be able to claim that for its newest medication the proportion of patients who experience side effects is less than 20%.

Q. What are the two possible conclusions (hypotheses) here?

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Hypothesis Tests• A statistical test is to check a statistical hypothesis

using data. It involves the five steps:

1. Set up the null (Ho) and alternative (H1) hypotheses

2. Find an appropriate test statistic (T.S.)

3. Find the rejection (critical) region (R.R.)

4. Reject Ho if the observed test statistic falls into R.R. and not reject Ho otherwise

5. Report the result in the context of the situation

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Determine Ho and H1• The null hypothesis Ho is the no-change hypothesis

• The alternative hypothesis H1 says that Ho is false

The Logic of Hypothesis Tests:

“Assume Ho is a possible truth until proven false”

Analogical to

“Presumed innocent until proven guilty”

The logic of the US judicial system

Q: What are the two possible conclusions?

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Determine Ho and H1Golden Rule: Ho must be a simple hypothesis.

Practical Rule: If possible, the hypothesis we hope to prove (called research hypothesis) goes to H1.

Back to the drug example, setting Ho and H1.

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Good!(Correct!)

H0 true H0 false

Type II Error, or “ββββ Error”

Type I Error, or

“αααα Error”

Good!(Correct)

we accept H0

we reject H0

Types of Errors

More Terms• α = Significance level of a test = Type I error rate

• Power of a test = 1-Type II error rate=1- β

• We only control α not β,

���� so we don’t say “accept Ho”.

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Report the Conclusion

• Reject Ho: the data shows strong evidence supporting Ha

Eg. The data shows strong evidence that the proportion of users who will experience side effects is less than 20% at significant level of 10%.

• Fail to reject Ho: the data does not provide sufficient evidence supporting Ha

Eg. Based on the data, there is not sufficient evidence to support the proportion is less than 20% at significant level of 5%.

Tests for One Mean

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Z Test

For normal populations or large samples (n >30)

And the computed value of Z is denoted by Z*.

n

XZ

/0

σµ−=

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Types of Tests

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Types of Tests

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Types of Tests

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Example 1 (Conti.)

Conduct a test for Ho: mu=2500 vs. H1: mu =3000 at 5% significant level.

1) What is the R.R.?

2) What is the power of the test?

Z test is the most powerful test!

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P-Values

• The p-value is the smallest level of significance to reject Ho at the observed value, also called the observed significance level.

p-value > α � fail to reject Ho

p-value < α � reject Ho (= accept Ha)

• That is, p-value is the probability of seeing as extreme as (or more extreme) what we observe, given Ho is true.

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P-Value• The level of significance (called α level) is usually

0.05

• p-value > α � fail to reject Ho (??)

• p-value < α � reject Ho (= accept Ha)

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Computing the p-Value for the Z-Test

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Computing the p-Value for the Z-Test

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Computing the p-Value for the Z-Test

P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)

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t Test• For normal populations with unknown σ

Eg. Revisit Example 1

)1(~/

0 −−= ntns

Xt

µ

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One Population

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Testing Hypotheses about a Proportion

• Three possible Ho and Ha

Ho Ha Type

p = po p = po Two-sided

p > po p < po One-sided (lower-tailed)

p < po p > po One-sided (upper-tailed)

Write them all as p=po in the future

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The z-test for a Proportion

• When 1) the sample is a random sample

2) n(po) and n(1-po) are both at least 10,

an appropriate test statistic for p is

n

pp

ppz

oo

o

)1(

ˆ

−−=

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Example: New Drug (Conti.)1. Ho: p > 20% vs. Ha: p < 20%

2. Z-test statistic; α = 0.053. Find rejection region or p-value

4. Decide if reject Ho or not

5. Report the conclusion in the context of the situation

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Hypothesis Test for the Difference of Two

Population Proportions

• Step 1. Set up hypotheses

Ho: p1 = p2 and three possible Ha’s:

Ha: p1 = p2 (two-tailed)

or

Ha: p1 < p2 (lower-tailed)

or

Ha: p1 > p2 (upper-tailed)

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Hypothesis Test for the Difference

between Two Population Proportions

• Step 2. calculate test statistic

where

+−

−=

21

21

11)ˆ1(ˆ

ˆˆ

nnpp

ppz

21

2211 ˆˆˆ

nn

pnpnp

++=

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Hypothesis Test for the Difference

between Two Population Proportions

• Step 3: Find p value

1. Must be two independent random samples; both are large samples:

And

2. When the above conditions are met, use Z-Table to find p-value.

• Steps 4 and 5 are the same as before

10)ˆ1(,ˆ 11 ≥− pnpn 10)ˆ1(,ˆ 22 ≥− pnpn

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Example: Bike to School

For 80 randomly selected men, 30 regularly bicycled to campus; while for 100 randomly selected women, 20 regularly bicycled to campus.

• Find the p-value for testing:

Ho: p1 = p2 vs. Ha: p1 > p2

Answer: z=2.60, p=0.0047

1: men; 2: women

Order Statistics• Min & Max

• Joint and other orders

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Order StatisticsProblem 1: Suppose X1, X2, …, X5 are a random sample from U[0,1]. Find the pdf of X(2).

Problem 2: Suppose X1, X2, …, Xn are a random sample from U[0,1]. Show that X(k) ~ beta(k,n-k+1).

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Order StatisticsThe CDF of X(k) and example

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Wilcoxon TestsHo: median of X = median of Y vs. H1: Ho is false

Wilcoxon tests (p. 448 - 450)

• assume the two distributions are of similar shapes but do not need to be normal

• See the supplementary material

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Exercise 8.5-9X = the life time of light bulb of brand A

Y = the life time of light bulb of brand B

Data: (in 100 hours)

X: 5.6 4.6 6.8 4.9 6.1 5.3 4.5 5.8 5.4 4.7

Y: 7.2 8.1 5.1 7.3 6.9 7.8 5.9 6.7 6.5 7.1

(a)Conduct the Wilcoxon test at 5 % level to test if brand B has longer life time in general.

A: W(Y)=145 > 128 or Z= 3.024 > 1.645; reject Ho

(a)Construct and interpret a Q-Q plot of these data.

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R Code for Q-Q Plot

> x<-c(5.6, 4.6, 6.8, 4.9, 6.1, 5.3, 4.5, 5.8, 5.4, 4.7)

> y<-c(7.2, 8.1, 5.1, 7.3, 6.9, 7.8, 5.9, 6.7, 6.5, 7.1)

> qqplot(x,y,xlab="life time of brand A", ylab="life time of brand B", main="qqplot of Life time of Brand A vs. Brand B")

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