10.1 DAY 2: Confidence

Intervals – The Basics

How Confidence Intervals Behave We select the confidence interval, and the

margin of error follows… We strive for HIGH confidence and a

SMALL margin of error. HIGH confidence says that our method almost

always gives correct answers. SMALL margin of error says that we have

pinned down the parameter quite precisely.

How Confidence Intervals Behave Consider margin of error…

The margin of error gets smaller when… z gets smaller. To accept a smaller margin of error,

you must be willing to accept lower confidence. σ gets smaller. The standard deviation σ measures

the variation in the population. n gets larger. We must take four times as many

observations in order to cut the margin of error in half.

nz

Ex 1: Video Screen Tension – Part 2

Suppose the manufacturer (from yesterday’s example) wants 99% confidence rather than 90%. The critical value for 99% confidence is z = 2.57. The 99% confidence interval for μ based on a SRS of 20 video terminals with mean x = 306.3 is:

)0.331,6.281(7.243.30620

4357.23.306

n

zx

Demanding 99% confidence instead of 90% confidence has increased the

margin of error from 15.8 to 24.7.

Sample Size for a Desired Margin of Error To determine the sample size that will yield

a confidence interval for a population mean with a specified margin of error, set the expression for the margin of error to be less than or equal to m and solve for n:

mn

z

Ex 2: How Many Monkeys?

Researchers would like to estimate the mean cholesterol level μ of a particular variety of monkey that is often used in lab experiments. They would like their estimate to be within 1 mg/dl of the true value of μ at a 95% confidence level. A previous study indicated that σ = 5 mg/dl. Obtaining monkeys is time-consuming and expensive, so researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate.

n

n

n

n

nz

04.96

8.9

)5)(96.1(

15

96.1

1

We must round up!!! We need 97

monkeys to estimate the

cholesterol levels to our satisfaction.

Ex 3: 2004 ElectionA poll taken immediately before the 2004 election showed that 51% of the sample intended to vote for John Kerry. The polling organization announced that they were 95% confident that the sample result was within + 2 points of the true percent of all voters who favored Kerry.

Ex 3: 2004 ElectionExplain in plain language to someone who knows no statistics what “95% confident” means in this announcement.

The method captures the unknown parameter 95% of the time.

The poll showed Kerry leading. Yet the organization said the election was too close to call. Explain.

Since the margin of error was 2%, the true value of p could be as low as 49%. Thus, the confidence interval contains some values of p, which suggests that Bush will win.

Ex 3: 2004 Election

On hearing the poll, a politician asked, “What is the probability that over half the voters prefer Kerry?” A statistician replied that this question can’t be answered from the poll results, and that it doesn’t even make sense to talk about such a probability. Explain.

First, the proportion of voters who favor Kerry is not random – either a majority favors Kerry or they don’t. Discussing probabilities has little meaning: the “probability” the politician asked about is either 1 or 0.

Some Cautions The size of the sample determines margin

of error. The size of the population does not influence the sample size.

The data must be a SRS from the population.

Different methods are needed for different designs (other than a SRS).

There is NO correct method for inference from data haphazardly collected with bias of unknown size.

…More Cautions Outliers can distort results. The shape of the population distribution

matters. When n>15, the confidence level is not greatly

disturbed by non-Normal populations unless extreme outliers or quite strong skewness are present.

So far, we have been given the standard deviation σ of the population. We will learn how to proceed with an unknown σ later.