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 

H o

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

.



p q o o n

The Reasoning of Hypothesis Testing

 Hypotheses  The null hypothesis • Translate our question into a statement about • model parameters Write

H o

:

parameter



value

 The alternative hypothesis • Contains the values of the parameter we accept if • we reject the null.

Write

H A

:

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



o

  

model”

The Reasoning of Hypothesis Testing

 Mechanics  Do the actual calculations of a test statistic from the data  The 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

 Conclusion  The 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 the statistic

z

 

p



p o

  

H o

:

p



p o

using . We use the hypothesized proportion to find the standard deviation, 

p q o o 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 P o  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