Confidence Interval Curve

Point Estimate vs Interval Estimate

Interval Estimate  An

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?

Confidence Interval A

 Use

an appropriate sample statistic:

• For the population mean, use the sample mean

• For the population proportion, use the sample proportion

confidence 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

•

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

A 95% Confidence Interval for a Population Proportion 

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

Confidence Interval A

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

Margin of Error  The

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

Finding the 95% Confidence Interval for a Population Proportion 



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

pˆ ± 1.96(standard errors)

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 =

This formula depends on the unknown population proportion, p In practice, we don’t know p, and we need to estimate the standard error

Finding the 95% Confidence Interval for a Population Proportion 

Finding the 95% Confidence Interval for a Population Proportion

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

pˆ ± 1.96(se), with se =

pˆ(1 - pˆ) n

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

p ˆ (1 − p ˆ) n

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

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

518 = 0.45 1154 (0.45)(0.55) se = = 0.015 1154 pˆ ± 1.96(se) = 1.96(0.015)

 Find

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

pˆ =

= 0.45 ± 0.03 = (0.42, 0.48)

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:

npˆ ≥ 15 and n(1 - pˆ) ≥ 15

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

“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

What is the Error Probability for the Confidence Interval Method?

How to Construct a Confidence Interval for a Population Mean  Point

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

How to Construct a Confidence Interval for a Population Mean 





How to Construct a Confidence Interval for a Population Mean  In

For large n…

•

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

and also

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

x ± z(

σ n

)

How to Construct a Confidence Interval for a Population Mean

practice, we don’t know the population standard deviation the sample standard deviation s for σ 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

Properties of the t-distribution  The

 In

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

Summary: 95% Confidence Interval for a Population Mean 

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

x±t ( .025



s ); df = n - 1 n

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

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?

To use this method, you need:

• •

Data obtained by randomization An approximately normal population distribution

Example: eBay Auctions of Palm Handheld Computers

Example: eBay Auctions of Palm Handheld Computers

 Consider

 “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

of selling prices for the two types of auctions:

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

Example: eBay Auctions of Palm Handheld Computers  To

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

Minimum Q1 Median Q3 178.00 221.25 240.00 246.75 210.00 225.00 235.00 250.00

Example: eBay Auctions of Palm Handheld Computers 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

Example: eBay Auctions of Palm Handheld Computers

 Let

Example: eBay Auctions of Palm Handheld Computers



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  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)

 The

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



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