Apr 20, 2023

Chapter 10: Chapter 10: Basics of Confidence IntervalsBasics of Confidence Intervals

In Chapter 10:

10.1 Introduction to Estimation

10.2 Confidence Interval for μ when σ is known

10.3 Sample Size Requirements

10.4 Relationship Between Hypothesis Testing and Confidence Intervals

§10.1: Introduction to EstimationTwo forms of estimation• Point estimation ≡ single best estimate of

parameter (e.g., x-bar is the point estimate of μ)• Interval estimation ≡ surrounding the point

estimate with a margin of error to create a range of values that seeks to capture the parameter; a confidence interval

Reasoning Behind a 95% Confidence Interval

• A schematic (next slide) of a sampling distribution of means based on repeated independent SRSs of n = 712 is taken from a population with unknown μ and σ = 40.

• Each sample derives a different point estimate and 95% confidence interval

• 95% of the confidence intervals will capture the value of μ

Confidence Intervals• To create a 95% confidence interval for μ,

surround each sample mean with a margin of error m that is equal to 2standard errors of the mean:

m ≈ 2×SE = 2×(σ/√n)

• The 95% confidence interval for μ is now

mx

This figure shows a sampling distribution of means.

Below the sampling distribution are five confidence intervals.

In this instance, all but the third confidence captured μ

Example: Rough Confidence Interval

Suppose body weights of 20-29-year-old males has unknown μ and σ = 40. I take an SRS of n = 712 from this population and calculate x-bar =183. Thus:

pounds 186 to1803183for CI 95%

35.122

5.1712

40

mx

SEmn

SE

x

x

Confidence Interval Formula

Here is a better formula for a (1−α)100% confidence interval for μ when σ is known:

Note that σ/√n is the SE of the mean

nzx

21

Confidence level

1 – α

Alpha level

α

Z value

z1–(α/2)

.90 .10 1.645

.95 .05 1.960

.99 .01 2.576

Common Levels of Confidence

90% Confidence Interval for μ

5.185 to5.180

5.2183712

40645.1183

for CI %9021.1

n

zx

Data: SRS, n = 712, σ = 40, x-bar = 183

95% Confidence Interval for μ

9.185 to1.180

9.2183712

40960.1183

for CI %95205.1

n

zx

Data: SRS, n = 712, σ = 40, x-bar = 183

99% Confidence Interval for μ

9.186 to1.179

9.3183712

40576.2183

for CI %99201.1

n

zx

Data: SRS, n = 712, σ = 40, x-bar = 183

Confidence Level and CI Length↑ confidence costs ↑ confidence interval length

Confidence level

Illustrative CI CI length = UCL – LCL

90% 180.5 to 185.5 185.5 – 180.5 = 5.0

95% 180.1 to 185.9 185.9 – 180.1 = 5.8

99% 179.1 to 186.9 186.9 – 179.1 = 7.8

10.3 Sample Size Requirements

2

1 2

m

zn

To derive a confidence interval for μ with margin of error m, study this many individuals:

Examples: Sample Size Requirements

Suppose we have a variable with = 15 and want a 95% confidence interval. Note, α = .05 z1–.05/2 = z.975 = 1.96

356.345

1596.1 use ,5For

22

1 2

m

znm

Smaller margins of error require larger sample sizes

1393.1385.2

1596.1 use ,5.2For

2

nm

8654.8641

1596.1 use ,1For

2

nm

round up to ensure precision

10.4 Relationship Between Hypothesis Testing and Confidence Intervals

A two-sided test will reject the null hypothesis at the α level of significance when the value of μ0 falls outside the (1−α)100% confidence interval

This illustration rejects H0: μ = 180 at α =.05 because 180 falls outside the 95% confidence interval.

It retains H0: μ = 180 at α = .01 because the 99% confidence interval captures 180.