section 7.2

Agresti/Franklin Statistics, 1 of 87

Section 7.2

How Can We Construct a Confidence Interval to Estimate a

Population Proportion?

Finding the 95% Confidence Interval for a Population Proportion

We symbolize a population proportion by p The point estimate of the population

proportion is the sample proportion We symbolize the sample proportion by p̂

A 95% confidence interval uses a margin of error = 1.96(standard errors)

[point estimate ± margin of error] =

errors) ard1.96(stand p̂

The exact standard error of a sample proportion equals:

This formula depends on the unknown population proportion, p

In practice, we don’t know p, and we need to estimate the standard error

pp )1(

In practice, we use an estimated standard error:

)ˆ1(ˆ

A 95% confidence interval for a population proportion p is:

)p̂-(1p̂ se with 1.96(se), p̂

Example: Would You Pay Higher Prices to Protect the Environment?

In 2000, the GSS asked: “Are you willing to pay much higher prices in order to protect the environment?”

• Of n = 1154 respondents, 518 were willing to do so

Find and interpret a 95% confidence interval for the population proportion of adult Americans willing to do so at the time of the survey

0.48) (0.42, 0.03 0.45

)015.0(96.1 1.96(se)p̂

015.01154

)55.0)(45.0(

45.01154

518p̂

Sample Size Needed for Large-Sample Confidence Interval for a Proportion

For the 95% confidence interval for a proportion p to be valid, you should have at least 15 successes and 15 failures:

15 )p̂-n(1 and 15 ˆ pn

“95% Confidence”

With probability 0.95, a sample proportion value occurs such that the confidence interval contains the population proportion, p

With probability 0.05, the method produces a confidence interval that misses p

How Can We Use Confidence Levels Other than 95%?

In practice, the confidence level 0.95 is the most common choice

But, some applications require greater confidence

To increase the chance of a correct inference, we use a larger confidence level, such as 0.99

A 99% Confidence Interval for p

2.58(se) ˆ p

Different Confidence Levels

In using confidence intervals, we must compromise between the desired margin of error and the desired confidence of a correct inference

• As the desired confidence level increases, the margin of error gets larger

What is the Error Probability for the Confidence Interval Method?

The general formula for the confidence interval for a population proportion is:

Sample proportion ± (z-score)(std. error)

which in symbols is

z(se) ˆ p

What is the Error Probability for the Confidence Interval Method?

Summary: Confidence Interval for a Population Proportion, p

A confidence interval for a population proportion p is:

)p̂-(1p̂z p̂

Summary: Effects of Confidence Level and Sample Size on Margin of Error

The margin of error for a confidence interval:

• Increases as the confidence level increases

• Decreases as the sample size increases

What Does It Mean to Say that We Have “95% Confidence”?

If we used the 95% confidence interval method to estimate many population proportions, then in the long run about 95% of those intervals would give correct results, containing the population proportion

Section 7.3

How Can We Construct a Confidence Interval To Estimate a

Population Mean?

How to Construct a Confidence Interval for a Population Mean

Point estimate ± margin of error The sample mean is the point

estimate of the population mean The exact standard error of the

sample mean is σ/ In practice, we estimate σ by the

sample standard deviation, s

For large n…• and also

For small n from an underlying population that is normal…

The confidence interval for the population mean is:

In practice, we don’t know the population standard deviation

Substituting the sample standard deviation s for σ to get se = s/ introduces extra error

To account for this increased error, we replace the z-score by a slightly larger score, the t-score

In practice, we estimate the standard error of the sample mean by se = s/

Then, we multiply se by a t-score from the t-distribution to get the margin of error for a confidence interval for the population mean

Properties of the t-distribution

The t-distribution is bell shaped and symmetric about 0

The probabilities depend on the degrees of freedom, df

The t-distribution has thicker tails and is more spread out than the standard normal distribution

t-Distribution

Summary: 95% Confidence Interval for a Population Mean

A 95% confidence interval for the population mean µ is:

To use this method, you need:• Data obtained by randomization

• An approximately normal population distribution

1-n df );( t.025

Example: eBay Auctions of Palm Handheld Computers

Do you tend to get a higher, or a lower, price if you give bidders the “buy-it-now” option?

Consider some data from sales of the Palm M515 PDA (personal digital assistant)

During the first week of May 2003, 25 of these handheld computers were auctioned off, 7 of which had the “buy-it-now” option

“Buy-it-now” option:

235 225 225 240 250 250 210

Bidding only:

250 249 255 200 199 240 228 255 232 246 210 178 246 240 245 225 246 225

Summary of selling prices for the two types of auctions:

buy_now N Mean StDev Minimum Q1 Median Q3

no 18 231.61 21.94 178.00 221.25 240.00 246.75 yes 7 233.57 14.64 210.00 225.00 235.00 250.00

buy_now Maximum

no 255.00

yes 250.00

To construct a confidence interval using the t-distribution, we must assume a random sample from an approximately normal population of selling prices

Let µ denote the population mean for the “buy-it-now” option

The estimate of µ is the sample mean:

x = $233.57 The sample standard deviation is:

s = $14.64

The 95% confidence interval for the “buy-it-now” option is:

which is 233.57 ± 13.54 or (220.03, 247.11)

64.14(44.257.233)(025.

The 95% confidence interval for the mean sales price for the bidding only option is:

(220.70, 242.52)

Notice that the two intervals overlap a great deal:

• “Buy-it-now”: (220.03, 247.11)

• Bidding only: (220.70, 242.52)

There is not enough information for us to conclude that one probability distribution clearly has a higher mean than the other

How Do We Find a t- Confidence Interval for Other Confidence Levels?

The 95% confidence interval uses t.025

since 95% of the probability falls between - t.025 and t.025

For 99% confidence, the error probability is 0.01 with 0.005 in each tail and the appropriate t-score is t.005

If the Population is Not Normal, is the Method “Robust”?

A basic assumption of the confidence interval using the t-distribution is that the population distribution is normal

Many variables have distributions that are far from normal

How problematic is it if we use the t- confidence interval even if the population distribution is not normal?

For large random samples, it’s not problematic

The Central Limit Theorem applies: for large n, the sampling distribution is bell-shaped even when the population is not

What about a confidence interval using the t-distribution when n is small?

Even if the population distribution is not normal, confidence intervals using t-scores usually work quite well

We say the t-distribution is a robust method in terms of the normality assumption

Cases Where the t- Confidence Interval Does Not Work

With binary data

With data that contain extreme outliers

The Standard Normal Distribution is the t-Distribution with df = ∞

The 2002 GSS asked: “What do you think is the ideal number of children in a family?”

The 497 females who responded had a median of 2, mean of 3.02, and standard deviation of 1.81. What is the point estimate of the population mean?

a. 497

c. 3.02

d. 1.81

section 7.2

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