# Confidence Interval Curve

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Interval Estimate An

Point Estimate vs Interval Estimate

interval estimate is an interval of numbers within which the parameter value is believed to fall.

Point Estimation: How Do We Make a Best Guess for a Population Parameter?A Use

Confidence Intervalconfidence interval is an interval containing the most believable values for a parameter The probability that this method produces an interval that contains the parameter is called the confidence level

For the population mean, use the sample For the population proportion, use thesample proportion mean

an appropriate sample statistic:

This is a number chosen to be close to 1, most commonly 0.95

A 95% Confidence Interval for a Population Proportion

Margin of Error The

Fact: Approximately 95% of a normal distribution falls within 1.96 standard deviations of the mean

That means:

With probability 0.95, the sample proportion falls within about 1.96 standard errors of the population proportion

margin of error measures how accurate the point estimate is likely to be in estimating a parameter The distance of 1.96 standard errors in the margin of error for a 95% confidence interval

Confidence IntervalA

Finding the 95% Confidence Interval for a Population Proportion

confidence interval is constructed by adding and subtracting a margin of error from a given point estimate When the sampling distribution is approximately normal, a 95% confidence interval has margin of error equal to 1.96 standard errors

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

p 1.96(standard errors)

Finding the 95% Confidence Interval for a Population Proportion

Finding the 95% Confidence Interval for a Population Proportion

The exact standard error of a sample proportion equals:

p(1 p) n

In practice, we use an estimated standard error: se = p (1 p ) n

This formula depends on the unknown population proportion, p In practice, we dont know p, and we need to estimate the standard error

Finding the 95% Confidence Interval for a Population Proportion

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

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

p 1.96(se), with se =

p(1 - p) n

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

Of n = 1154 respondents, 518 werewilling to do so

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

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

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

518 = 0.45 1154 (0.45)(0.55) se = = 0.015 1154 p 1.96(se) = 1.96(0.015) p= = 0.45 0.03 = (0.42, 0.48)

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

95% Confidence With

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

np 15 and n(1 - p) 15

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

Different Confidence Levels In

What is the Error Probability for the Confidence Interval Method?

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

How to Construct a Confidence Interval for a Population Mean Point

How Can We Construct a Confidence Interval To Estimate a Population Mean?

estimate margin of error sample mean is the point estimate of the population mean The exact standard error of the sample mean is / n In practice, we estimate by the sample standard deviation, s The

How to Construct a Confidence Interval for a Population Mean

How to Construct a Confidence Interval for a Population Mean In

For large n

and also

For small n from an underlying population that is normal The confidence interval for the population mean is:

x z(

n

)

practice, we dont know the population standard deviation the sample standard deviation s for to get se = s/ n introduces extra error To account for this increased error, we replace the z-score by a slightly larger score, the t-score Substituting

How to Construct a Confidence Interval for a Population Mean In

Properties of the t-distribution The

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

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

Summary: 95% Confidence Interval for a Population Mean

Example: eBay Auctions of Palm Handheld Computers Do

A 95% confidence interval for the population mean is:

xt (.025

s ); df = n - 1 n

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

To use this method, you need:

Data obtained by randomization An approximately normal population distribution

Example: eBay Auctions of Palm Handheld Computers Consider

Example: eBay Auctions of Palm Handheld Computers Buy-it-now

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

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

Example: eBay Auctions of Palm Handheld Computers Summary

Example: eBay Auctions of Palm Handheld Computers To

of selling prices for the two types of auctions:Minimum Q1 Median Q3 178.00 221.25 240.00 246.75 210.00 225.00 235.00 250.00

buy_now N Mean StDev no 18 231.61 21.94 yes 7 233.57 14.64 buy_now Maximum no 255.00 yes 250.00

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

Example: eBay Auctions of Palm Handheld Computers Let

Example: eBay Auctions of Palm Handheld Computers

denote the population mean for the buy-it-now option The estimate of is the sample mean: x = $233.57 The sample standard deviation is: s = $14.64

The 95% confidence interval for the buy-itnow option is:

x t.025 (

s 14.64 ) = 233.57 2.44( ) n 7

which is 233.57 13.54 or (220.03, 247. 11)

Example: eBay Auctions of Palm Handheld Computers The

Example: eBay Auctions of Palm Handheld Computers Notice

95% confidence interval for the mean sales price for the bidding only option is: (220.70, 242.52)

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

For