Chapter 20:Testing Hypotheses about Proportions

“Half the money I spend on advertising is wasted; the trouble is I don’t know which half.”

-John Wanamaker

Hypotheses

Hypothesis: a supposition a proposition or principle which is supposed or

taken for granted, in order to draw a conclusion or inference for proof of the point in question

In statistical tests of hypotheses, we assume that a hypothesis is true. If the data are consistent with the hypothesis,

we retain the hypothesis If the data are not consistent with the

hypothesis, we reject the hypothesis

Testing Hypotheses

The Null Hypothesis The starting hypothesis The null hypothesis specifies a population

model parameter of interest and proposes a value for the parameter

Standard Deviation (not Standard Error) We assume the null hypothesis to be true, so

we have a value for the model parameter p.

oH

o op qSD p

n

The Reasoning of Hypothesis Testing Hypotheses

The null hypothesis

• Translate our question into a statement about

model parameters

• Write

The alternative hypothesis

• Contains the values of the parameter we accept if

we reject the null.

• Write

:oH parameter value

:AH parameter value

The Reasoning of Hypothesis Testing

Plan Specify the model you will use to test the null

hypothesis and the parameter of interest All models require assumptions, so state them and

check any corresponding conditions Include the name of the test you plan to perform End with a statement such as

“Because the conditions are satisfied, it id appropriate

to model the sampling distribution of the proportion with a

model” ,oN p SD p

The Reasoning of Hypothesis Testing

MechanicsDo the actual calculations of a test statistic

from the dataThe ultimate goal is to obtain a P-value

• The probability that the observed statistic value could occur if the null model were correct

• If the P-value is small enough, reject the null hypothesis

The Reasoning of Hypothesis Testing

ConclusionThe conclusion in a hypothesis test is

always a statement about the null hypothesis

“reject” or “fail to reject” the null hypothesis

Consider the size of the effect by examining a confidence interval

One-proportion z-test

The conditions for the one-proportion z-test are the same as for the one-proportion z-interval.

We test the hypothesis : using

the statistic . We use the

hypothesized proportion to find

the standard deviation,

o o

o

o o

H p p

p pz

SD p

p qSD p

n

.

When the conditions are

met and the null

hypothesis is true, the

statistic follows the

standard Normal model,

so we can use that

model to obtain a P-

value.

Alternative Alternatives

Two-sided alternative

When we are equally

interested in

proportions that deviate

from p in either

direction

Alternative Alternatives

One-sided alternative An alternative hypothesis

that focuses on deviations from the null hypothesis value in only one direction.

The P-value is the probability of deviating only in the direction of the alternative away from the null hypothesis value

TI-83+ Tips

STAT TESTS 5: 1-Prop ZTest

TI-83+ Tips

Specify the hypothesized

proportion Po

Enter x, the observed number

of wins

Specify the sample size

Decide on one- or two-tailed

test

Calculate

P-Values and Decisions:What to Tell About a Hypothesis Test

Hypothesis tests are useful when making a decision

It is a good idea to report the confidence interval for the parameter of interest

The P-value is highly context-dependent The importance of the issue also factors into the

choice of P-value The conclusion about any null hypothesis should

be accompanied by the P-value of the test. If possible, it should also include a confidence interval for the parameter of interest