Introduction to Inference

Confidence Intervals

Statistical Inference

Provides a method for drawing conclusions about a population from sample data.

We use probability to express the strength of our

conclusions.

Remember!

The purpose of this chapter is to describe the reasoning used in statistical inference, not the methods.

Later chapters will show how to modify these techniques to make them practically useful.

SAT Math Scores in NC

Suppose the SAT Math test is administered to an SRS of 500 North Carolina high school seniors and the mean score turns out to be 461. Suppose (unrealistically) we also know the standard deviation of the population is 100. What can we say about the mean score in the population of all 100,000 North Carolina high school seniors?

Statistical Confidence

68-95-99.7 Rule

In 95% of all samples, the unknown µ (the population mean score) lies between and .

€

x −9

€

x + 9

Conclusion

Proper Conclusion: We are 95% confident that the unknown true mean score for NC seniors is between 452 and 470. (If we selected many different SRS of size 500, 95% of the samples taken would yield an interval containing µ.)

WRONG Conclusion: There is a 95% chance that µ falls between 452 and 470.

Confidence Interval

We just created a 95% confidence interval!

Form: estimate ± margin of error

It gives the probability that the interval will capture the true parameter value in repeated samples.

Confidence Interval

http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/index.html

Conditions

The construction of a confidence interval for a population mean µ is appropriate when

the data come from an SRS from the population of interest

the sampling distribution of is

approximately normal

€

x

More Confidence Intervals

A level C

confidence interval for µ is

The bottom row in Table C gives the values of z* for many confidence levels

€

x ± z *σ

n

Inference Toolbox

1. Identify the population of interest and the parameter you want to draw conclusions about.

2. Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

3. If the conditions are met, carry out the inference procedure.

4. Interpret your results in the context of the problem.

Corn Yield(Homework #10.8 on p 550)

a) Find the 90% confidence interval for the mean yield µ for this variety of corn.

b) Find the 95% confidence interval.c) Find the 99% confidence interval.d) How do the margins of error in a),

b), and c) change as the confidence level increases?

More Corn(Homework #10.9 on p 551)

a) Compute the 95% confidence interval for the mean yield µ.

b) Is the margin of error larger or smaller for the sample of 60 plots than the margin of error we found for the sample of 15 plots?

c) Will the 90% and 99% intervals for a sample of size 60 be wider or narrower than those for n=15?

CI Behavior

margin of error =

The margin of error gets smaller when:

z* gets smaller (smaller confidence level)

gets smaller n gets larger

€

z *σ

n

Choosing Your Sample Size

A wise user of statistics never plans data collection without planning the inference at the same time!

You can arrange to have both high confidence and a small margin of error by taking enough observations.

Solve for n.

€

z *σ

n≤ m

The Moral of the Story

Think of a confidence interval as a net and the parameter as a fish. Our ultimate goal is to catch the fish using the smallest net necessary. If our net is too small, it will be very hard to catch the fish. (Our confidence level is too low.) If our net is too big, it’s hard to use. (Our margin of error is too big.) In this chapter, we’ll see how to create the right-sized net for the situation.