7-7-11

7-7-22

Chapter Seven

Confidence Intervals

McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

這一章完全靠背公式套之

7-7-33

Confidence Intervals

7.1 Large Sample Confidence Intervals for a Population Mean

7.2 Small Sample Confidence Intervals for a Population Mean

7.3 Sample Size Determination

7.4 Confidence Intervals for a Population Proportion

*7.5 A Comparison of Confidence Intervals and Tolerance Intervals

7-7-44

7.1 Large Sample Confidence Intervals for a Mean

n

s/2zx

If is unknown and the sample size is large (n 30), estimate by the sample standard deviation s and a )100% confidence interval for is

If the sampled population is normal or the sample size is large (n 30), a )100% confidence interval for is

n

/2zx

(1-a)100%=confidence level

背公式套之

7-7-55

Example: Large Sample Confidence Interval for Mean

Example 7.1: Gas Mileage Case

49

7992.01.9605531.31zx 0.025

n

s95% Confidence Interval

99% Confidence Interval]78.31,33.31[2238.05531.31

49

7992.02.5755531.31zx 0.005

n

s

]85.31,26.31[2940.05531.31

背公式套之

7-7-66

Three 95.44Confidence Intervals for

Gas Mileage Example best coverage

7-7-77

The Effect of on Confidence Interval Width

7-7-88

7.2 Small Sample Confidence Intervals for a Mean

n

s/2tx

If the sampled population is normally distributed with mean , then a )100% confidence interval for is

is the t point giving a right-hand tail area of /2 under the t curve having n – 1 degrees of freedom.

/2t

t 分配 : small sample is the key word

7-7-99

Example: Small Sample Confidence Interval for Mean

Example 7.4: Debt-to-Equity Ratios

15

1921.02.1453433.1tx 0.025

n

s

1.4497] [1.2369, 0.10643433.1

查表 t

7-7-1010

Effect of Degrees of Freedom on the t-distribution

As the number of degrees of freedom increases, the spread of the t distribution decreases and the t curve approaches the standard normal curve.

df makes difference

7-7-1111

7.3 Sample Size Determination

A sample size

n =z

B/2

2

will yield an estimate of , precise to within B units of , with 100(1-) confidence.

x

背公式套之 ,從信賴區間而來 ,誤差界線 , 民調用之 , 1068

7-7-1212

7.4 Confidence Intervals for a Population Proportion

1

)ˆ1(ˆzp̂ /2

n

pp

If the sample size n is large , then a )100% confidence interval for p is

]5both)ˆ1(andˆ[ pnpn

背公式套之； Bernouli trial, condition is above, based on central limit theory

n

7-7-1313

Example: Confidence Interval for a Proportion

Example 7.8: Phe-Mycin side effects

p =35

200=.175

1200

)825.0)(175.0(960.10.175

1

)ˆ1(ˆzp̂ /2

n

pp

]228.0,122.0[053.00.175

7-7-1414

Determining Sample Size for Confidence Interval for p

A sample size2

/2

B

zp)-p(1=n

will yield an estimate , precise to within B units of p, with 100(1-) confidence.

p̂

Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p.

背公式套之

7-7-1515

*7.5 A Comparison of Confidence Intervals and Tolerance Intervals

7-7-1616

Summary: Selecting an Appropriate Confidence Interval for a Population Mean

7-7-1717

Confidence Intervals

Summary: 7.1 Large Sample Confidence Intervals

for a Population Mean

7.2 Small Sample Confidence Intervals for a Population Mean

7.3 Sample Size Determination

7.4 Confidence Intervals for a Population Proportion

*7.5 A Comparison of Confidence Intervals and Tolerance Intervals