AP Statistics Section 11.1 BMore on Significance Tests

Conditions for Significance TestsThe three conditions that should be satisfied before we conduct a hypothesis test about an unknown population mean or proportion are the same as they were for confidence intervals: 1. _______ from the population of interest. 2. Distribution of must be ________________

For : _________________________________For : ________________________

3. _________________________If sampling w/o replacement ___________

SRSp̂ and x Normal approx.

30)(n CLTor Normal population xp̂ 10p)-n(1 and 10np

nsobservatiot Independen10nN

Example 1: Check that the conditions from the paramedic example in section 11.1 A are met. SRS: Normality of : Independence:

x

400 of SRS

Normal approx. is that dist. a gives CLT so 400n

4000or 10(400)calls all of pop.

sot replacemen without Calls

Test StatisticsA significance test uses data in the form of a test statistic. The following principles apply to most tests: (1) the test statistic compares the value of the parameter as stated in the __ to an estimate of the parameter from the sample data. (2) values of the estimate far from the parameter value in the direction specified by the alternativehypothesis give evidence _____________ (3) to assess how far the estimate is from the parameter, standardize the estimate.

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0Hagainst

In many common situations, the test statistic has the form:

test statistic = ----------------------------------------- valueedhypothesiz - valuesampledist. sample theofdeviation standard

4002

7.648.6 z

?zWhy

deviation. standard population theknow We

Because the result is over two standard deviations below the hypothesized mean 6.7, it gives good evidence that the mean RT this year is not equal to 6.7 minutes,

but rather, less than 6.7 minutes.

The probability, computed assuming __________, that the observed sample

outcome would take a value as extreme as or more extreme than that

actually observed is called the __________ of the test.

trueis 0H

value- p

The smaller the P-value is, the stronger the evidence is against

provided by the data.

Example 3: Let’s go back to our paramedic example. The P-value is the probability of getting a sample

result at least as extreme as the one we did ( = 6.48) if were true. In other words, the P-value is

calculated assuming . We just found the z-score for this exact situation, so using

Table A or our calculator, this P-value is _______. So if is true, and the mean RT this year is still 6.7

minutes, there is about a _____ chance that the city manager would obtain a sample of 400 calls with a mean RT of 6.48 minutes or less. The small P-value

provides strong evidence against and in favor of the alternative minutes.

x67:0 H

)48.6( xP 7.6

0139.0H

%4.1

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67: aH

If the Ha is two-sided, both directions count when figuring the

P-value.

Example 4: Suppose we know that differences in job satisfaction scores in Example 3 of section 11.1 A follow a Normal

distribution with standard deviation . If there is no difference in job satisfaction between the two work

environments, the mean is _______. Thus H0: ________. The Ha says simply “there is a difference,” thus Ha:________. Data from 18 workers gave 17. That is, these

workers preferred the self-paced environment on average. Find the p-value for this situation and interpret it.

600

00

20.1

1860

017

z

1151.

.23022(.1151) value-P

A p-value of .2302 indicates that 23.02% of the time we will get a sample where is at least as big as 17 when . An outcome that would occur this often when is

not good evidence that .

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00

Statistical SignificanceWe can compare the P-value with a fixed value

that we regard as decisive. This amounts to announcing in advance how much evidence

against we will insist on. The decisive value of P is called the significance level. We write it as

____, the Greek letter alpha.

If the P-value , we say that the data are

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levelat t significanlly statistica

Example 5: Back to the paramedic example. We found the P = 0.0139. The

result is statistically significant at the .05 level since P < ____ but it is not significant

at the .01 level since P > ____

“Significant” in the statistical sense does not mean “_____________.” It means

simply “not likely to happen just by _________.”

05.

01.

important

chance

Interpreting Results in ContextThe final step in performing a significance

test is to draw a conclusion about the competing claims you were testing. As with

confidence intervals, your conclusion should have a clear connection to your calculations and should be stated in the context of the problem. These are called

the 3 C’s.

In significance testing, there are two accepted methods for drawing

conclusions:

In examples 3 and 4 of this section we simply stated the p-value and

interpreted it in the context of the problem.

In example 5, we went on to determine if the data was statistically

significant be comparing our P-value to our significance level . We can either _______ or _______________ the Ho

based on whether our result is statistically significant at a given

significance level.

reject reject tofail

Warning: if you are going to draw a conclusion based on statistical

significance, then the significance level should be stated before the

data are produced.

Example 6: For the paramedic example, we calculated the P-value to be 0.0139. If we were using an significance level, we

would _____ minutes ( ________ ) since ______ ( __________ ). It appears that the mean response time to all life-

threatening calls this year is less than last year’s average of 6.7 minutes ( ______ ).

05.

reject 7.6:0 H conclusion.05p connection

context

Finally, stating a P-value is more informative than simply giving a “reject” or “fail to reject”

conclusion at a given significance level. For example, a P-value of 0.0139 allows us to

reject at the level. But the P-value, 0.0139 gives us a better sense of how strong the

evidence against is. The P-value is the smallest level at which the data are significant.

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