1. Introduction 2. Statistical Confidence

Statistics 11/Economics 40

Lecture 14

Confidence Intervals (6.1)

1. Introduction

In Chapter 5, the parameters are known, and we estimate the chance of various sample outcomes. In

Chapter 6 the parameters are unknown, and we draw conclusions from sample outcomes to make

educated guesses about the parameters.

Rather than calculating exact probabilities, we use statements of confidence to express the strength of our

conclusions (6.1). Later, in 6.2, we will test hypotheses that make assumptions about the parameters and

allow us to calculate probabilities just like we do in Chapter 5.

2. Statistical Confidence

A CONFIDENCE INTERVAL is a range of values (i.e. values derived from sample information) that we

think covers the true parameter and we state our confidence in the sample outcome with a percentage. An

example might be (handout) "Stock Market Confidence Shaky" (they are referring to a different kind of

confidence, but they also use confidence intervals in their article)

The article states that 601 likely voters were surveyed and 48% answered "somewhat confident" to the

question "how confident are you that the stock market will remain stable". A "margin of error" of +/4.0% is given.

You have probably heard polling information stated in this way before: 48% with a margin of error of 4%

or 48% plus or minus 4%. The margin of error is often stated to give you a sense of how accurate the

statistic is and how confident you are about it.

This is a confidence interval for the population percentage (that is, if we could ask EVERYONE the

question "how confident are you that the stock market will remain stable") or you could think of it as a

statement of confidence that this sample "covers" the true parameter. The confidence interval is calculated

from the sample percentage and sample standard deviation. Up until now, we have been in a situation

where we know exactly what the population parameters are, now we do not, but we have samples and can

make statements of confidence about our samples.

Take a long look at Figure 6.2 in your text (p. 438). (see handout)

Things to keep in mind:

(1) In about 68% of all samples, the sample percentage will be within one standard deviation of the

true population percentage.

(2) In about 95% of all samples, the sample percentage will be within about two standard deviations

(Z=1.96) of the population percentage.

From the poll, we would say that we were 95% confident that the true percentage of likely voters

nationwide who are somewhat confident that the stock market will remain stable is covered by the

interval from 44% to 52%. Conventionally, the margin of error stated reflects about two standard

deviations.

(3) In about 99% of all samples, the sample percentage will be within Z=2.576 standard deviations of

the population percentage.

NOTES: You can never be 100% confident. There is always the chance that you could have generated a

sample that just by chance (not anything you did wrong) is nowhere near the parameter. Sample theory

(see Chapter 5 again) tells us that some percentage of samples will always be far away from the

parameter, even though the procedure used to select it was random. Also remember a natural property of

the normal curve, it never crosses or touches the x-axis, so even at 10 S.D. there is a non-zero chance that

your confidence interval will not cover the parameter, but the chance of that happening is very small.

Statistics 11/Economics 40

Lecture 14

Confidence Intervals (6.1)

And remember -- if you generate a bad sample , e.g. biased, non-random, your statistics will be bad and

while you can generate a confidence interval, it's meaningless.

3. Constructing Confidence Intervals

Constructing a confidence interval for a population parameter involves five steps:

A. Find the sample statistic of interest. This is our ESTIMATE of the population parameter. Let's

look at another example: 52% of Americans have been on a blind date according to a survey of

1,004 Americans.

B. Compute the standard deviation for the sample distribution; for simple random samples involving

percentages, the standard deviation is:

(.52 )(.48 )

= .016

1004

C. Then choose the level of confidence you are interested in from a normal table using the area

percentages. Use the associated Z as a multiplier, let's just use 2.0 to get a 95% confidence

interval (this is a standard approximation, using a Z=1.96 would be more correct)

so .016 * 2 = .032 and then multiply by 100 to make it a percentage or .032*100 = 3.2%

D. Add and subtract the result in C from the result in A so, 52% +/- 3.2%. This is your "margin of

error" that is, how accurate you believe your statistic is based on the variability of the estimate.

4. Notes on Confidence

a. A typical confidence interval has the form "estimated value, plus or minus Z times the SD of the

sample distribution". In other words, take a statistic calculated from a sample and add and subtract some

margin of error (the Standard deviation multiplied by some value of Z).

b. If the original population is normally distributed with a known standard deviation, or if the sample

size is "large", then the distribution of the sample statistic is normal, and using Z from the normal table is

appropriate. (If the original distribution is normal with an unknown standard deviation, or if it is not

normal and the sample is small, you will not use Z, more on this later.)

c. Your margin of error will depend on the choice of a confidence level. A lower confidence will give

you a smaller margin of error. A higher confidence will give you a larger margin of error.

d. If your standard deviation is small, it is easier to get a more precise fix on the parameter. Your

margin of error is smaller for populations with smaller standard deviations.

e. If your n increases in size, it will reduce your margin of error. If your n gets smaller, it will

increase your margin of error. Therefore, you can adjust your sample size to accommodate a desired

margin of error (see page 443).

f. If the sample is known to be biased, the confidence interval can be calculated, but it is worthless.

5. Interpreting Confidence

1. The best interpretation for a confidence interval is as follows: "We did a procedure of drawing a

sample, computing a statistic, standard deviation, etc. This procedure will give us a correct interval X% of

the time and an incorrect interval 100-X% of the time. We hope this is one of the correct times. Thus, for

about X% of all samples, the interval "sample statistic + or - [Z*(standard deviation)]" covers the true

population percentage.

Statistics 11/Economics 40

Lecture 14

Confidence Intervals (6.1)

2. It is not correct to talk about the chance a particular confidence interval contains the parameter. For

example, you should not say "there is a X% chance that the parameter is in the confidence interval"

because these confidence intervals vary with each sample and the parameter never varies (the

parameter is fixed and unchanging).

Any single confidence interval either covers the true parameter or it does not. Examine the article

from Forbes Magazine.

3. Another way you might think about this. When you KNOW the TRUE POPULATION

PARAMETER, you can make a statement like: there is a 95% CHANCE that the SAMPLE STATISTIC

will be in the range of the parameter plus or minus two standard deviations. (this is Chapter 5)

Example: if you know the parameter is ¨¬ =40 and the standard deviation of the sampling distribution

is 2.5%, then there is a 95% chance that the sample average will be in the range of 40 plus or minus 5.

But when you DO NOT KNOW THE TRUE POPULATION PARAMETER, you are forced to make

statements like this: I am 95% confident that the POPULATION PARAMETER is in the range of the

statistic plus or minus two standard deviations.

Example: if you don't know the parameter and the sample statistic is 40 and the SD is 2.5, then you

are 95% confident that the parameter is covered by the range of 40 plus or minus 5.

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