Hypothesis Tests with Proportions

Chapter 10

Write down the first number that you think of for the following . . .

Pick a two-digit number between 10 and 50, where both digits are ODD and the digits do not repeat.

•What possible values fit this description?

•Record your answer on the dotplot on the board.

•What do you notice about this distribution?

•Did you expect this to happen?

• What proportion of the time would I expect to get the value 37 if the values were equally likely to occur?

• Is the difference in these proportions significant?

How do I know if this p-hat is significantly different from the

1/8 that I expectexpect to happen?

A hypothesis test will help me decide!

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 2: (Back to the opening activity) Is the proportion of students who answered 37 higher than the expected proportion of 1/8?

Where p is the true proportion of people who answered “37”

H0: p = 1/8

Ha: p > 1/8

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 be NOT equal!

Level of Significance Level of Significance ActivityActivity

P-value -P-value -

•Assuming H0 is true, the probability that the statistic would have a value as as extreme or moreextreme or more than what is actually observed

Notice that this is a conditional probability

The statistic is our p-hat!

Why not find the probability that the p-hat equals a certain value?

Remember that in continuous distributions, we cannot find

probabilities of a single value!

P-values -P-values -

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

is the probability of getting values more (or

less) than our p-hat?p̂p̂

We can use normalcdf to find this probability.

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 –• Our statistic (p-hat) is statistically

significant if the p-value is as smallas small or smaller smaller than the level of significance ().

Decisions:• If p-value < , “rejectreject” the null

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

hypothesis at the level.

Our “guilty” verdict.Our “not guilty” verdict.

Remember that the verdict is never “innocent” – so we can

never decide that the null is true!

Facts about p-values:• ALWAYS make the 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!

Calculating p-values• For z-test statistic (z) –

–Use normalcdf(lb,ub) to find the probability of the test statistic or more extreme

–Remember the standard normal curve is comprised of z’s where = 0 and = 1

We will see how to compute this value

tomorrow.

Since we are in the standard normal curve,

we do not need here.

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

2) two-tail test z = -2.4

Normalcdf(1.6,∞)

P-value = .0548

Normalcdf(-∞,-2.4) × 2

P-value = .0164

z

z

Double the p-value since this is a two-tailed

test!

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

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 - parameter

Test statisticSD of parameter

z n

pp

pp

1

ˆ p̂ pp ˆ

Let’s put all the steps together!

Example 2 revisited: Is the proportion of people who think of the value 37 significantly higher than what we expect? Use = 0.05.

What confidence level would be equivalent to this right-tailed test with = 0.05?

Calculate this confidence interval.

How do the results from the confidence interval compare to the results of the hypothesis test?

Example 5: A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?

Assumptions:

•Have an SRS of people

•np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal.

•Population of people is at least 4000.

H0: p = .2 where p is the true proportion of people who

Ha: p > .2 heard the ad

05.1056.25.1

400)8(.2.

2.225.

valuepz

Since the p-value > , I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2. The company will not renew their advertising contract with the radio station.

Use the parameter in the null hypothesis to check assumptions!

Use the parameter in the null hypothesis to calculate standard

deviation!

Calculate the appropriate confidence interval for the above problem.

= .225 + .041 = (.184, .266)

How do the results from the confidence interval compare to the results of the hypothesis test?

The confidence interval contains the parameter of .2 thus providing no evidence that more than 20% had heard the ad.

* ( )(1 ) .225*.775.225 1.96

phat phatp hat z

n n