Objective: To test claims about inferences for means, under specific conditions - PowerPoint Presentation

Slide1

Objective: To test claims about inferences for means, under specific conditions

Hypotheses tests for means

Chapter 23 – Part BSlide2

The hypotheses for proportions are similar to those for

proportions.

In fact, they are the same. We use

µ instead of p. H0: µ = 3.2

HypothesesSlide3

There are four basic parts to a hypothesis test:

Hypotheses

Model

MechanicsConclusion

Let’s look at these parts in detail…

Reasoning of Hypothesis testingSlide4

1. Hypotheses

The null hypothesis:

To perform a hypothesis test, we must first translate our question of interest into a statement about model parameters.

In general, we have H0

:

parameter = hypothesized value.

The alternative hypothesis:

The alternative hypothesis, HA, contains the values of the parameter we consider plausible when we reject the null.

Reasoning of Hypothesis testing (cont.)Slide5

Model

To

plan a statistical hypothesis test, specify the

model you will use to test the null hypothesis and the parameter of interest.All models require assumptions, so state the assumptions and check any corresponding conditions.

Your

conditions should conclude

with a statement

such as:

Because the conditions are satisfied, I can model the sampling distribution of the proportion with a Normal model.

Watch out, though. It might be the case that your model step ends with “Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case, stop and reconsider (proceed with caution)

Reasoning of Hypothesis testing (cont.)Slide6

Model (cont.)

Don’t forget to name your test!

The test about

means is

called a

one-sample t-test

.

Reasoning of Hypothesis testing (cont.)Slide7

Model (cont.)

One-sample

t-test for the meanThe conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval.

We test the hypothesis

H

0

:  = 0

using the

statistic

The

standard error

of the sample mean is When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t model with n – 1 df. We use that model to obtain a P-value.

Reasoning of Hypothesis testing (cont.)Slide8

Model (cont.)

Finding the P-Value

Either use the table provided, or you may use your calculator:

normalcdf( is used for z-scores (if you know

)

tcdf

( is used for critical t-values (when you use s to estimate

)2nd

 Distribution

tcdf(lower bound, upper bound, degrees of freedom)

Reasoning of Hypothesis testing (cont.)Slide9

Mechanics

Under “mechanics” we place the actual calculation of our test statistic from the data.

Different tests will have different formulas and different test statistics.

Usually, the mechanics are handled by a statistics program or calculator, but it’s good to know the formulas.

Reasoning of Hypothesis testing (cont.)Slide10

Mechanics (continued)

The

ultimate goal of the calculation is to obtain a

P-value.The P-value is the probability that the observed statistic value (or an even more extreme value) could occur if the null model were correct.

If

the P-value is small enough, we’ll reject the null hypothesis.

Note

: The P-value is a conditional probability—it’s the probability that the observed results could have happened

if the null hypothesis is true

.Reasoning of Hypothesis testing (cont.)Slide11

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.

And, as always, the conclusion should be stated in context.

Reasoning of Hypothesis testing (cont.)Slide12

Check Conditions and show that you have checked these!

Random Sample

: Can we assume this?

10% Condition: Do you believe that your sample size is less than 10% of the population size?Nearly Normal: If you have raw data, graph a histogram to check to see if it is approximately symmetric and sketch the histogram on your paper.

If you do not have raw data, check to see if the problem states that the distribution is approximately Normal.

Steps for Hypothesis testing for one-sample t-test for meansSlide13

State the test you are about to conduct

Ex) One-Sample t-Test for Means

Set up your hypotheses

H0: HA:

Calculate your test statistic

Draw a picture of your desired area under the t-model, and calculate your P-value.

Steps for Hypothesis testing for

one-sample

t-test for

means (

cont.)Slide14

Make your conclusion.

When

your P-value is small enough (or below

α, if given), reject the null hypothesis. When your P-value is not small enough, fail to reject the null hypothesis.

Steps for Hypothesis testing for

one-sample

t-test for

means (cont.)Slide15

Given a set of data:

Enter data into L1

Set up STATPLOT to create a histogram to check the nearly Normal condition

STAT  TESTS  2:T-Test Choose Stored Data, then specify your data list (usually L1)Enter the mean of the null model and indicate where the data are (>, <, or

)

Given

sample mean and standard deviation:

STAT  TESTS  2:T-Test

Choose Stats  enter

Specify the hypothesized mean and sample statisticsSpecify the tail (>, <, or )Calculate

Calculator tipsSlide16

A company has set a goal of developing a battery that lasts over 5 hours (300 minutes) in continuous use. A first test of 12 of these batteries measured the following lifespans (in minutes):

321,

295

, 332, 351, 281, 336, 311, 253, 270

,

326

,

311, and 288. Is there evidence that the company has met its goal? 

     Example 1Slide17

Find

a 90% confidence interval for the mean lifespan of this type of battery. 

     Example 1 (continued)Slide18

Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by 10 tasters for one new cola recipe:

Are these data good evidence that the cola lost sweetness?

Example 2 (Partners)Slide19

Day 2Slide20

Psychology experiments sometimes involve testing the ability of rats to navigate mazes. The mazes are classified according to difficulty, as measured by the mean length of time it takes rats to find the food at the end. One researcher needs a maze that will take the rats an average of about one minutes to solve. He tests one maze on several rats, collecting the data shown. Test the hypothesis that the mean completion time for this maze is 60 seconds at an alpha level of 0.05. What is your conclusion?

Example 3

38.4

57.6

46.2

55.5

62.5

49.5

38.0

40.9

62.8

44.3

33.9

93.8

50.4

47.9

35.0

69.2

52.8

46.2

60.1

56.3

55.1Slide21

Confidence intervals and hypothesis tests are built from the same calculations.

They have the same assumptions and conditions.

You can approximate a hypothesis test by examining a confidence interval.

Just ask whether the null hypothesis value is consistent with a confidence interval for the parameter at the corresponding confidence level.

Confidence intervals & Hypothesis testsSlide22

Because confidence intervals are two-sided, they correspond to two-sided tests.

In general, a confidence interval with a confidence level of

C

% corresponds to a two-sided hypothesis test with an -level of

100

–

C

%.The relationship between confidence intervals and one-sided

hypothesis tests

is a little more complicated.A confidence interval with a confidence level of C% corresponds to a one-sided hypothesis test with an -level of

½(100 –

C

)%.

Confidence intervals & Hypothesis tests (cont.)Slide23

Here’s some shocking news for you: nobody’s perfect. Even with lots of evidence we can still make the wrong decision.

When we perform a hypothesis test, we can make mistakes in

two

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

(

Type I error)

II. The

null hypothesis is false, but we fail to reject it.

(Type II error)Making ErrorsSlide24

Which type of error is more serious depends on the situation at hand. In other words, the

importance of the error is context dependent.

Here’s an illustration of the four situations in a hypothesis test:

Making Errors (cont.)Slide25

http://www.youtube.com/watch?v=Q7fZXEW4mpA

What type of error was made?

How about OJ Simpson?

Making Errors (cont.)Slide26

How often will a Type I error occur?

A

Type I error is rejecting a true null

hypothesis. To reject the null hypothesis, the P-value must fall below . Therefore, when the null is true, that happens exactly with a probability of . Thus, the

probability of a Type I error is our



level

.When H

0

is false and we reject it, we have done the right thing.A test’s ability to detect a false null hypothesis is called the power of the test.Making Errors (cont.)Slide27

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.It’s harder to assess the value of 

because we don’t know what the value of the parameter really is.

When the null hypothesis is true, it specifies a single parameter value,

H

0: parameter = hypothesized value.When the null hypothesis is false, we do not have a specific parameter; we have many possible values.

There is no single value for

 --we can think of a whole collection of ’s, one for each incorrect parameter value.

Making Errors (cont.)Slide28

One way to focus our attention on a particular

 is to think about

the

effect size. Ask “How big a difference would matter?”We could reduce



for

all

alternative parameter values by increasing .This would reduce 

but increase the chance of a Type I error.

This tension between Type I and Type II errors is inevitable.The only way to reduce both types of errors is to collect more data. Otherwise, we just wind up trading off one kind of error against the other.

Making Errors (cont.)Slide29

The

power of a test is the probability that it correctly rejects a false null hypothesis.

The

power of a test is 1 –  ; because  is the probability that a test fails

to reject a false null hypothesis and power is the probability that it does reject

.

Whenever

a study fails to reject its null hypothesis, the test’s power comes into question.

When we calculate power, we imagine that the null hypothesis is false.

Power of the testSlide30

The value of the power depends on how far the truth lies from the null hypothesis value.

The distance between the null hypothesis value,



0 , and the truth, 

,

is called the

effect size.

Power depends directly on effect size

. It is easier to see larger effects, so the farther  is from 0, the greater the power.

Power of the test (cont.)Slide31

Day 1

: pp. 499-503 #

1 - 5

pp. 554-559 # 23, 24 Day 2: pp. 554-559 #

1cd, 2cd, 29, 30, 33, 35

Day 3

:

pp. 554-559 # 22, 25 – 28, 34Assignments