1

Chapter 9: Introduction to Inference

2

Thumbtack Activity

Toss your thumbtack in the air and record whether it lands either point up (U) or point down (D). Do this 25 times (n=25). Calculate p-hat.

Repeat the above process two more times, for a total of three estimates. Record your p-hat on a separate post-it

note.

3

We’ve just begun a sampling distribution.

Strictly speaking, a sampling distribution is: A theoretical distribution of the values of a

statistic (in our case, the proportion) in all possible samples of the same size (n=25 here) from the same population.

Sampling Variability: The value of a statistic varies from sample-

to-sample in repeated random sampling. We do not expect to get the same exact

value for the statistic for each sample!

4

Definitions Parameter:

A number that describes the population of interest. Rarely do we know its value, because we do not

(typically) have all values of all individuals from a population.

We use µ and σ for the mean and standard deviation of a population.

P and σp for proportions.

Statistic: A number that describes a sample. We often use a

statistic to estimate an unknown parameter. We use x-bar and s for the mean and standard

deviation of a sample. P-hat and σp-hat for proportions.

5

Sampling Distribution

The sampling distribution answers the question, “What would happen if we repeated the sample or experiment many times?” Formal statistical inference is based

on the sampling distribution of statistics.

6

Inference

Inference is the statistical process by which we use information collected from a sample to infer something about the population of interest.

Two main types of inference: Interval estimation (Section 9.1) Tests of significance (Section 9.2)

7

Constructing Confidence Intervals

Back to the thumbtack activity … Interpretation of 95% C.I.:

If the sampling distribution is approximately normal, then the 68-95-99.7 rule tells us that about 95% of all p-hat values will be within two standard deviations of p (upon repeated samplings). If p-hat is within two standard deviations of p, then p is within two standard deviations of p-hat. So about 95% of the time, the confidence interval will contain the true population parameter p.

8

Internet Demonstration, C.I.

http://bcs.whfreeman.com/yates/pages/bcs-main.asp?s=00020&n=99000&i=99020.01&v=category&o=&ns=0&uid=0&rau=0

9

Interpretation of 95% CI (Commit to memory!)

95% of all confidence intervals constructed in the same manner will contain the true population parameter. 5% of the time they will not.

10

11

p. 492

12

13

Finding a 95% C.I.

14

Practice

See example 9.3, p. 495 Exercises 9.1-9.4, p. 495

15

Creating the C.I.

Estimate +/- Margin of error

16

Another practice problem

9.5, p. 496

17

p. 496

18

Finding a confidence interval, general form

19

Figure 9.5, p. 502

20

21

Practice

9.9 and 9.10, p. 505

22

Confidence intervals with the calculator

23

9.2 Significance Testing

An evolutionary psychologist at Harvard University claims that 80% (p=0.80) of American adults believes in the theory of evolution. To test his claim, he takes an SRS of 1,120 adults. Here are the results:

851 said “Yes” when asked, “Do you believe in the theory of evolution?”

What is the proportion who said yes? Is this enough evidence to say that the proportion of

adults who do not believe in the theory of evolution is different from 0.80?

24

Example, cont.

This requires a significance test: Hypotheses:

Ho: p=0.80 Ha: p≠0.80

Let’s use our calculators to conduct the appropriate test: 5: 1-prop ztest

25

Example Results

P-value

26

p. 516

27

Hypotheses

Alternate hypothesis Ha:

Can be one-sided (Ha: p> some number or p< some number)or two-sided (Ha: p≠ some number)

28

HW

9.24-9.26, p. 521 Reading: pp. 509-525

29

p. 519

30

Sampling Applet

http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/