Chapter 21

What Is a Confidence Interval?

Chapter 15 1

Chapter 21 2

Recall from previous chapters:Parameter

fixed, unknown number that describes the population

Statisticknown value calculated from a samplea statistic is used to estimate a parameter

Sampling Variabilitydifferent samples from the same population may yield different values of the sample statisticestimates from samples will be closer to the true values in the population if the samples are larger.

Chapter 21 3

The Rule for Sample Proportions

Chapter 21 4

Rule Conditions and IllustrationFor rule to be valid, must have

1. Random sample2. ‘Large’ sample size

Chapter 21 5

Case Study: FingerprintsA Question

Assume that the proportion of all men

who have leftward asymmetry is 15%.

Is it unusual to observe a sample of 66 men with a sample

proportion ( ) of 30% if the true population proportion (p) is 15%?

p̂

Chapter 21 6

Case Study: FingerprintsAnswer to Question

Where should about 95% of the sample proportions lie?

mean plus or minus two standard deviations0.15 2(0.044) = 0.0620.15 + 2(0.044) = 0.238

95% should fall between 0.062 & 0.238

Chapter 21 7

Formula for a 95% Confidence Interval

• sample proportion plus or minus two standard deviations ofthe sample proportion:

€

p± 2p(1− p)

n

Chapter 21 8

n

ppp

)ˆ1(ˆ2ˆ

standard error (estimated standard deviation of )p̂

Formula for a 95% Confidence Interval

Chapter 21 9

Margin of Error

nn

n

pp

15.015.0

ˆ1ˆ

2

2

)(

)( (plus or minus part of C.I.)

Chapter 21 10

Formula for a C-level (%) Confidence Interval for the Population Proportion

n

ppp z )ˆ1(ˆˆ *

where z* is the critical value of the standard normal distribution for confidence level C

Chapter 21 11

Common Values of z*Confidence Level

C Critical Value

z* 50% 0.67

60% 0.84

68% 1

70% 1.04

80% 1.28

90% 1.64

95% 1.96 (or 2)

99% 2.58

99.7% 3

99.9% 3.29

Gambling on Sports

Chapter 21 12

A December 2007 Gallup Poll con-sisting of a random sample of 1027 adult Americans found that 17% had gambled on sports in the last 12 months. Find a 95% confidence interval for the proportion of all adult Americans who gambled on sports in this time period. How would you interpret this interval?

A 99% confidence interval

Chapter 15 13

The BRFSS random sample of 2166 college graduates in California in 2006 found that 279 had engaged in binge drinking in the past year. We want a 99% confidence interval for the proportion p of all college graduates in California who engaged in binge drinking in the past year.

Chapter 15 14

Our statistical software has a “random number generator” that is supposed to produce numbers scattered at random between 0 to 1. If this is true, the numbers generated come from a population with μ = 0.5. A command to generate 100 random numbers gives outcomes with mean x = 0.536 and s = 0.312. Give a 90% confidence interval for the mean of all numbers produced by the software.

Chapter 15 15

We would like to estimate the mean GPA of the population of all WSU undergraduates. We would like to compute a 90 percent confidence interval for this mean, and we would like our interval to have margin of error no greater than 0.1. The Registrar's Office says that a reliable guess of the population standard deviation is 0.5. What is the smallest number of students we should recruit to achieve our goals?