Chapter 20 Testing Hypothesis about proportions Example:

Metal Manufacturer Ingots 20% defective (cracks)

After Changes in the casting process: 400 ingots and only 17% defective

Is this a result of natural sampling variability or there is a reduction in the cracking rate?

Hypotheses We begin by assuming that a

hypothesis is true (as a jury trial). Data consistent with the hypothesis:

Retain Hypothesis Data inconsistent with the hypothesis:

We ask whether they are unlikely beyond reasonable doubt.

If the results seem consistent with what we would expect from natural sampling variability we will retain the hypothesis. But if the probability of seeing results like our data is really low, we reject the hypothesis.

Testing Hypotheses

Null Hypothesis H0

Specifies a population model parameter of interest and proposes a value for this parameter

Usually: No change from traditional value No effect No difference

In our example H0:p=0.20 How likely is it to get 0.17 from sample

variation?

Testing Hypotheses (cont.) Normal Sampling distribution

How likely is to observe a value at least 1.5 standard deviations below the mean of a normal model

Management must decide whether an event that would happen 6.7% of the time by chance is strong enough evidence to conclude that the true cracking proportion has decreased

02.0400

80.020.0)ˆ(

npq

pSD 17.0ˆ p

5.102.020.017.0 z

067.0)5.1( zP

A Trial as a Hypothesis Test The jury’s null hypothesis is

H0 : innocent

If the evidence is too unlikely given this assumption, the jury rejects the null hypothesis and finds the defendant guilty. But if there is insufficient evidence to convict the defendant, the jury does not decide that H0 is true and declare him innocent. Juries can only fail to reject the null hypothesis and declare the defendant “not guilty”

The Reasoning of Hypothesis Testing Hypothesis

To perform a hypothesis test, we must specify an alternative hypotheses. Remember we can never prove a null hypothesis, only reject it or retain it. If we reject it, we then accept the alternative

Example: Pepsi or Coke p : proportion preferring coke H0 : p = 0.50 HA : p ≠ 0.50

The Reasoning of Hypothesis Testing (cont.) Plan

Specify the model and test you will use (proportions, means).

We call this test about the value of a proportion a one-proportion z-test

Mechanics Actual Calculation of a test from the data. P-value : the probability that the observed

statistic value could occur if the null model were correct. If the P-value is small enough, we reject the null hypothesis

The Reasoning of Hypothesis Testing (cont.)

Conclusion The conclusion in a hypothesis test is

always a statement about the null hypothesis. The conclusion must state either that we reject or that we fail to reject the null hypothesis

Alternatives Two-sided Alternative

HA : p ≠ 0.50 (Pepsi – Coke) The P-value is the

probability of deviating in either direction from the null hypothesis

One-sided Alternative H0 : p = 0 HA : p < 0.20 (Ingots) The P-value is the

probability of deviating only in the direction of the alternative away from the null hypothesis value.

Exercises

Page 467 #1 #3 #20

Chapter 21More About Tests Example : Therapeutic Touch (TT)

One-proportion z-test 15 TT practitioners 10 trials each H0 : p=0.50 HA : p>0.50 Random Sampling Independence 10% condition Success/Failure condition Observed proportion 0.467 Find the P-value…

How to Think About P-values

A P-value is a conditional probability. It is the probability of the observed statistic given that the null hypothesis is true.

P-value : P(Observed statistic value|H0)

Alpha Levels When the P-value is small, it tells us that

our data are rare given the null hypothesis. We can define a “rare event” arbitrarily by

setting a threshold for our P-value. If our P-value falls below that point we’ll

reject the null hypothesis. We call such results “statistically significant” the threshold is called an alpha level or significance level.

Alpha Levels (cont.) = 0.10 = 0.05 = 0.01

Rejection Region

One Sided Two sided

Making Errors Type I error

The null hypothesis is true, but we mistakenly reject it.

Type II error The null hypothesis is false but we fail to reject

it.

Types of errors

Examples Medical disease testing

I : False Positive II : False Negative

Jury Trial I : Convicting an innocent II : Absolving someone guilty

Probabilities of errors

To reject H0, the P-value must fail below . When H0 is true that happens exactly with probability so when you choose the level , you are setting the probability of a Type I error to .

When H0 is false and we fail to reject it, we have made a Type II error. We assign the letter to the probability of this mistake

Power The power of a test is the probability that it

correctly rejects a false null hypothesis. When the power is high, we can be confident that we’ve looked hard enough.

We know that is the probability that a test fails to reject a false null hypothesis, so the power of the test is the complement 1 -

When we calculate power, we have to imagine that the null hypothesis is false. The value of the power depends on how far the truth lies from the null hypothesis value. We call this distance between the null hypothesis value p0 and the truth p the effect size.

Chapter 22Comparing Two Proportions Recall (Ch.16)

The variance of the sum or difference of two independent random quantities is the sum of their individual variances

Example of the cereals

)()()( YVarXVarYXVar

)()()( YVarXVarYXSD

Comparing Two Proportions (cont.) The Standard Deviation of the

Difference Between Two Proportions

For proportions from the data

2

22

1

1121 )ˆˆ(

nqp

nqp

ppSD

2

22

1

1121

ˆˆˆˆ)ˆˆ(nqp

nqp

ppSE

Assumptions and Conditions Random Sampling 10% condition Independent Samples Condition

The two groups we are comparing must also be independent of each other (usually evident from the way the data is collected).

Example : Same group of people before and after a

treatment are not independent Success and failure condition in each

sample

The Sampling Distribution

The sampling distribution for a difference between two independent proportions Provided the assumptions and

conditions the sampling distribution of is modeled by a normal model with mean and standard deviation

21 ˆˆ pp

21 pp

2

22

1

1121 )ˆˆ(

n

qp

n

qpppSD

A two-proportion z-interval When the conditions are met, we are ready to

find the confidence interval for the difference of two proportions p1-p2. Using the standard error of the difference

The interval is

The critical value z* depends on the particular confidence level.

2

22

1

1121

ˆˆˆˆ)ˆˆ.(.

n

qp

n

qpppES

)ˆˆ.(.*ˆˆ 2121 ppESzpp

Exercises

Two-proportion z-interval(page 493, 496)

Example Snoring

Random sample of 1010 Adults From 995 respondents:

37% snored at least few nights a week Splitting in two age categories:

Under 30 Over 30 26.1% of 184 39.2% of 811

Is the difference of 13.1% real or due only to sampling variability?

Example (cont. snoring) H0 : p1 – p2 = 0

But p1 and p2 are linked from H0

p1 = p2

Pooling: Combining the counts to get an overall

proportion

2

22

1

1121

ˆˆˆˆ)ˆˆ.(.

n

qp

n

qpppES

21

21ˆnnSuccessSuccess

ppooled

3678.0995366

81118431848

ˆ pooledp

222

111

ˆ

ˆ

pnSuccess

pnSuccess

Two-Proportion z-test The conditions for the two-proportion z-

test are the same as for the two-proportion z-interval . We are testing the hypothesis:

H0 : p1 = p2

Because we hypothesize that the proportions are equal, we pool them to find

And we use the pooled value to estimate the standard error

21

21ˆnnSuccessSuccess

ppooled

2121

ˆˆˆˆ)ˆˆ(..

n

qp

n

qpppES pooledpooledpooledpooled

pooled

Two-Proportion z-test (cont.)

Now we find the test statistic using the statistic

When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value

)ˆˆ(..ˆˆ

21

21

ppESpp

zpooled

Example (cont. snoring) Randomization 10% Condition Independent samples condition Success / Failure

The P-value is the probability of observing a difference greater or equal to 0.131

The two sided P-value is 0.0008. This is rare enough, so we reject the null hypothesis and conclude that there us a difference in the snoring rate between this two age groups.

0394.0)6322.0)(3678.0()6322.0)(3678.0(

)ˆˆ(..21

21 nn

ppES pooled

131.0261.0392.0ˆˆ 21 pp

33.30394.0

0131.0 z

Exercise

Page 508 #16

Homework #5

Page 423 #8, 16 Page 443 #12, 18 Page 467 #2, 4, 6, 12 Page 491 #20