AP Exam Prep: Essential Notes

Chapter 11: Inferencefor Distributions

11.1 Inference for Means of a Population11.2 Comparing Two Means

Moving away from z … In chapter 10, when we knew σ, we calculated

a z-score for a particular mean as follows:

n

xz

/

Now, we do not know σ, so we calculate a t-score, which provides somewhat of a “fudge-factor” because we do not know σ, but must estimate it from the sample :

ns

xt

/

Standard error

of the mean

One-sample t-procedures (p. 622)

Confidence interval:

n

stx *

Hypothesis test:

ns

xt

/0

In both cases, σ is unknown.

Matched Pairs t Procedures Matched pairs designs: subjects are matched

in pairs and each treatment is given to one subject in the pair (randomly).

One type of matched pairs design is to have a group of subjects serve as their own pair-mate. Each subject then gets both treatments (randomize the order).

Apply one-sample t-procedures to the observed differences.

Example 11.4, p. 629 Note H0

Look at Figure 11.7, p. 631

Conditions for Inference about a Mean (p. 617)

SRS Observations from the population

have a normal distribution with mean µ and standard deviation σ. Symmetric and single-peaked

essential.

Using t-procedures

See Box, p. 636 SRS very important! n<15: do not use t-procedures if the data are

clearly non-normal or if outliers are present. n at least 15: t-procedures can be used

except in the presence of outliers or strong skewness.

n at least 40: t-procedures can be used for even clearly skewed distributions.

By CLT

11.2 Comparing Two Means

The goal of two-sample inference problems is to compare the responses of two treatments or to compare the characteristics of two populations.

We must have a separate sample from each treatment or each population.

Unlike the matched-pairs designs.

A two-sample problem can arise from a randomized comparative experiment that randomly divides subjects into two groups and exposes each group to a different treatment.

Conditions for Significance TestsComparing Two Means (p. 650)

Two SRSs from distinct populations. Samples are independent (matching violates this

assumption). We measure the same variable for each sample.

Both populations are normally distributed. Means and standard deviations of both are unknown.

Two-sample t-test

The appropriate t-statistic is as follows. The degrees of freedom calculation is complex; we will use our calculators to provide this for us (the df are usually not whole numbers for two-sample tests).

2

22

1

21

2121 )()(

ns

ns

xxt

=0 for the H0:µ1=µ2

Two-sample confidence intervalfor µ1-µ2

Draw an SRS of size n1 from a normal population with unknown mean µ1, and draw an independent SRS of size n2 from a normal population with unknown mean µ2. The confidence interval for µ1-µ2 is given by the following:

2

22

1

21*

21 )(n

s

n

stxx

Again, we need the df for t*, but we will let the calculator do that for us.

Using t-procedures for two-sample analyses

See Box, p. 636 SRS very important! n1+n2<15: do not use t-procedures if the

data are clearly non-normal or if outliers are present.

n1+n2 at least 15: t-procedures can be used except in the presence of outliers or strong skewness.

n1+n2 at least 40: t-procedures can be used for even clearly skewed distributions.

By CLT

Chapter 12: Inference for Proportions

12.1 Inference for a Population Proportion12.2 Comparing Two Proportions

Conditions for Inference abouta Proportion (p. 687)

SRS N at least 10n For a significance test of H0:p=p0:

The sample size n is so large that both np0 and n(1-p0) are at least 10.

For a confidence interval: n is so large that both the count of successes, n*p-

hat, and the count of failures, n(1 - p-hat), are at least 10.

Normal Sampling Distribution

If these conditions are met, the distribution of p-hat is approximately normal, and we can use the z-statistic:

npp

ppz

)1(

^

Inference for a Population Proportion

Confidence Interval:

Significance test of H0: p=p0:

n

ppzp

)1(^^

*^

npp

ppz

)1( 00

0

^

Choosing a Sample Size (p. 695)

Our guess p* can be from a pilot study, or we could use the most conservative guess of p*=0.5.

Solve for n. Example 12.9, p. 696.

mn

ppZ

)1( ***

Conditions: Confidence Intervals for Comparing Two Proportions

SRS from each population N>10n All of these are at least 5:

)1(

)1(

2

^

2

2

^

2

1

^

1

1

^

1

pn

pn

pn

pn

Calculating a Confidence Interval for Comparing Two Proportions (p. 704)

2

2

^

2

^

1

1

^

1

^

*

2

^

1

^ )1()1()(

nnz

pppppp

Significance Tests forComparing Two Proportions

The test statistic is:

21

21^

nn

XXp

21

^^

^

2

^

1

11)1(

nnpp

ppz

Where,

Conditions: Significance Test for Comparing Two Proportions

SRS from each population N>10n All of these are at least 5:

)1(

)1(

2

^

2

2

^

2

1

^

1

1

^

1

pn

pn

pn

pn

Chapter 13: Chi-Square Procedures

13.1 Test for Goodness of Fit

13.2 Inference for Two-Way Tables

M&Ms Example

Sometimes we want to examine the distribution of proportions in a single population.

As opposed to comparing distributions from two populations, as in Chapter 12.

Does the distribution of colors in your bags match up with expected values?

We can use a chi-square goodness of fit test. Χ2

We would not want to do multiple one-proportion z-tests.

Why?

Performing a X2 Test

1. H0: the color distribution of our M&Ms is as advertised:

Pbrown=0.30, Pyellow=Pred=0.20, and Porange=Pgreen=Pblue=0.10

Ha: the color distribution of our M&Ms is not as advertised.

2. Conditions:1. All individual expected counts are at least 1.2. No more than 20% of expected counts are less than 5.

3. Chi-square statistic: EEOX /)( 22

Section 13.2 (Two-way tables)

Example 13.4, pp. 744-748

Is there a difference between proportion of successes?

At left is a two-way table for use in studying this question.

Explanatory Variable: Type of Treatment

Response Variable: Proportion of no relapses

Relapse?

Treatment No Yes Total

Desipramine

14 10 24

Lithium 6 18 24

Placebo 4 20 24

Total 24 48 72

Expected Counts and Conditions

All expected counts are at least 1, no more than 20% less than 5.

totaltable

alcolumn tot totalrowcount Expected

Chapter 14: Inference about the Model

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R2 = 0.9454

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Confidence Intervals for the Regression Slope (p. 788)

If we repeated our sampling and computed another model, would we expect a and b to be exactly the same?

Of course not, given what we’ve learned about random variation and sampling error!

We are interested in the true slope (β), which is unknowable, but we are able to estimate it.

Confidence Interval for the slope β of the true regression line:

bxay ^

dfnt

SEtb b

)2(*

*

Given in output from stats package.

Is β=0?

H0: β=0 vs. Ha: β ≠0 or β>0 or β<0

Perform a t-test:

bb SE

b

SE

bt

0