Hypothesis Tests with Proportions Chapter 10 Notes: Page 169.

Hypothesis Tests with Proportions

Chapter 10

Notes: Page 169

What are hypothesis What are hypothesis tests?tests?

Calculations that tell us if the sample statistics (p-hat) occurs by random chance or not OR . . . if it is statistically significantIs it . . .

– a random occurrence due to natural variation?

– an occurrence due to some other reason?

Statistically significant means that it is NOTNOT a random

chance occurrence!

Is it one of the sample

proportions that are likely to

occur?Is it one that isn’t likely to

occur?

These calculations (called the test statistictest statistic) will tell

us how many standard deviations a sample

proportion is from the population proportion!

Nature of hypothesis tests Nature of hypothesis tests --•First begin by supposing the

“effect” is NOT present•Next, see if data provides

evidence against the supposition

Example: murder trial

How does a murder trial work?

First - assume that the person is innocentThen – mustmust have

sufficient evidence to prove guilty

Hmmmmm …Hypothesis tests use the same process!

Steps:Steps:

1) Assumptions2) Hypothesis statements &

define parameters3) Calculations4) Conclusion, in context

Notice the steps are the same as a confidence interval except we add

hypothesis statements – which you will learn

today

Assumptions for z-test:Assumptions for z-test:

• Have an SRS of context• Distribution is (approximately)

normal because both np > 10 and n(1-p) > 10

• Population is at least 10n

YEA YEA –These are the same

assumptions as confidence intervals!!

Check assumptions for the Check assumptions for the following:following:Example 1: A countywide water conservation campaign was conducted in a particular county. A month later, a random sample of 500 homes was selected and water usage was recorded for each home. The county supervisors wanted to know whether their data supported the claim that fewer than 30% of the households in the county reduced water consumption after the conservation campaign.

•Given SRS of homesGiven SRS of homes•Distribution is approximately Distribution is approximately normal because np=150 & n(1-normal because np=150 & n(1-p)=350 (both are greater than 10)p)=350 (both are greater than 10)•There are at least 5000 homes in There are at least 5000 homes in the county.the county.

How to write hypothesis How to write hypothesis statementsstatements• Null hypothesis – is the statement

(claim) being tested; this is a statement of “no effect” or “no difference”

• Alternative hypothesis – is the statement that we suspect is true

HH00::

HHaa::

How to write How to write hypotheses:hypotheses:Null hypothesis H0: parameter = hypothesized value

Alternative hypothesis Ha: parameter > hypothesized value

Ha: parameter < hypothesized value

Ha: parameter = hypothesized value

Example 3: A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. Is this claim too high?

Where p is the true proportion of vaccinated people who do not get the flu

H0: p = .7

Ha: p < .7

Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses :Where is the

true mean amperage of the fuses

H0: = 40

Ha: = 40

Facts to remember about Facts to remember about hypotheses:hypotheses:• Hypotheses ALWAYS refer to

populations (use parameters – never statistics)

• The alternative hypothesis should be what you are trying to prove!

• ALWAYS define your parameter in context!

Activity: For each pair of hypotheses, indicate which are not legitimate & explain why

1H1H e)

6H4H d)

1H1H c)

123H123H b)

15H15H a)

a0

a0

a0

a0

a0

.ˆ:;.ˆ:

.:;.:

.:;.:

:;:

:;:

pp

pp

xx

Must use parameter (population) x is a statistics

(sample) is the

population proportion!Must use same

number as H0!P-hat is a statistic – Not a parameter!

Must NOT be equal!

P-values -P-values -

•Assuming H0 is true, the probability that the statistic would have a value as as extreme or moreextreme or more than what is actually observedIn other words . . . is it

far out in the tails of the distribution?

Level of Significance Level of Significance ActivityActivity

Level of significance -Level of significance - • Is the amount of evidence

necessary before we begin to doubt that the null hypothesis is true

• Is the probability that we will reject the null hypothesis, assuming that it is true

• Denoted by – Can be any value– Usual values: 0.1, 0.05, 0.01– Most common is 0.05

Statistically significant –

• The p-value is as smallas small or smaller smaller than the level of significance ()

• If p-value > , “fail to rejectfail to reject” the null hypothesis at the level.

• If p-value < , “rejectreject” the null hypothesis at the level.

Facts about p-values:• ALWAYS make decision about the

null hypothesis!• Large p-values show support for

the null hypothesis, but never that it is true!

• Small p-values show support that the null is not true.

• Double the p-value for two-tail (=) tests

• Never acceptNever accept the null hypothesis!

Never “accept” the null hypothesis!

Never “accept” the null hypothesis!

Never “accept” the null hypothesis!

At an level of .05, would you reject or fail to reject H0

for the given p-values?

a) .03b) .15c) .45d) .023

Reject

Reject

Fail to reject

Fail to reject

Calculating p-values

•For z-test statistic ––Use normalcdf(lb,ub)

–Remember that z’s form the standard normal curve with = 0 and = 1

Draw & shade a curve & calculate the p-value:1) right-tail test z = 1.6

2) two-tail test z = -2.4

P-value = 1-.9452 =.0548

P-value =.0082 + (1-.9918)

=.0082 + .0082

=.0164-2.4 +2.4

Writing Conclusions:

1) A statement of the decision being made (reject or fail to reject H0) & why (linkage)

2) A statement of the results in context. (state in terms of Ha)

AND

“Since the p-value < (>) , I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.” Be sure to write Ha

in context (words)!

Example 3 revisited: A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. In a test, vaccinated people were exposed to the flu. The test statistic for the results is z = -1.38. Is this claim too high? Write the hypotheses, calculate the p-value & write the appropriate conclusion for = 0.05.

H0: p = .7Ha: p < .7Where p is the true proportion of vaccinated people who get the flu

P-value = normalcdf(-10^99,-1.38) =.0838

Since the p-value > , I fail to reject H0. There is not sufficient evidence to suggest that the proportion of vaccinated people who do not get the flu is less than 70%.

Formula for hypothesis test:Formula for hypothesis test:

statistic of SD

parameter - statisticstatisticTest

z n

pp

pp

1

ˆ p̂ pp ˆ

Homework:

Page 172 and 173