Chapter 8 Review

Chapter 8 Confidence Interval Estimation Chapter Topics State Estimation Process Introduce Properties of Point Estimates Explain Confidence Interval Estimates Compute Confidence Interval Estimation for Population Mean ( known and unknown) • Compute Confidence Interval Estimation for Population Proportion • Compute Sample Size Statistical Methods • • • •

Statistical Methods Descriptive Statistics

Inferential Statistics

Estimation Inference Process

Estimates & Tests

Hypothesis Testing

Population

Sample Statistics X, Ps

Sample

Population Parameters Are Estimated with Point Estimator

Estimate Population Parameter... Mean µ

with Sample Statistic X

Proportion

p

ps

Variance

σ2

s

µ1 − µ2

X1 − X 2

Differences

2

Confidence Interval Estimation Process

Population

Random Sample

Mean, µ, is unknown

Mean X = 50

I am 95% confident that µ is between 40 & 60.

Confidence Interval Estimates Provide range of values – Takes into consideration variation in sample statistics from sample to sample – Is based on observation from one sample – Gives information about closeness to unknown population parameters – Is stated in terms of level of confidence Never 100% certain Elements of Confidence Interval Estimation

A Probability That the Population Parameter Falls Somewhere Within the Interval. Confidence Interval

Confidence Limit (Lower)

Sample Statistic

Confidence Limit (Upper)

Confidence Intervals

X ± Z ⋅σ X = X ± Z ⋅ σ n

µ − 2.58⋅σ X

µ −1.645⋅σ X

µ −1.96⋅σ X

σx_ µ µ +1.645⋅σ X

µ + 2.58⋅σ X

X

µ +1.96⋅σ X 90% Samples 95% Samples 99% Samples

Level of Confidence • Probability that the unknown population parameter falls within the interval • Denoted (1 - =% )α level of confidence α Is Probability That the Parameter Is Not Within the Interval • The typical values are 99%, 95%, 90%. Corresponding Z values are 2.58, 1.96, 1.645 (for two side problems). Experiment: throw 2 dice, use mean ±2 to construct a 94.5% confidence interval estimate for µ. Interval and Level of Confidence

Sampling Distribution of the Mean

µ − Zα / 2σ X Intervals extend from

α /2

σX 1−α

µX = µ

X − Zσ X

X

( 1 − α ) 100%

of intervals constructed contain µ ; α 100% do not.

to

X + Zσ X

µ + Zα / 2σ X

α /2

Confidence Intervals

Factors Affecting Interval Width 1. Data Dispersion Measured by σ 2. Sample Size

X  / n 3. Level of Confidence (1 - α) Affects Z Confidence Interval Estimates

Confidence Intervals Mean

σ Known

Proportion

σ Unknown

Confidence Interval Estimate For Mean (σ Known) Assumptions – Population standard deviation (σ) is known – Population is normally distributed – If not normal, can be approximated by normal distribution (n ≥ 30) Confidence Interval Estimate

σ σ X − Zα / 2 ⋅ ≤ µ ≤ X + Zα / 2 ⋅ n n

Example :

Population is normally distributed with σ = 10. The mean of a random sample of n = 25 is X = 50. Set up a 95% confidence interval estimate for µ.

X − Zα ⋅ σ ≤ µ ≤ X + Zα ⋅ σ 2 n 2 n 50−1.96⋅ 10 ≤ µ ≤ 50+1.96⋅ 10 25 25 46.08 ≤ µ ≤ 53.92 Confidence Interval Estimate Mean (σ Unknown) Assumptions Population Standard Deviation (σ) Is Unknown Population Must Be Normally Distributed Use Student’s t Distribution Confidence Interval Estimate:

X − t α / 2, n −1 ⋅

S n

≤ µ ≤ X + t α / 2, n −1 ⋅

S n

Student’s t Table

Upper Tail Area df

.25

.10

.05

α/2

Assume: n = 3 df = n - 1 = 2 α = .10 α/2 =.05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

α/2 =.05

3 0.765 1.638 2.353

t Values

0 2.920

t

Confidence Interval Estimate for Proportion Assumptions Two categorical outcomes Population follows binomial distribution Normal approximation can be used if n·p ≥ 5 and n·(1 - p) ≥ 5 Confidence interval estimate:

p S − Zα / 2

pS ( 1 − pS ) n

≤ p ≤ pS + Zα / 2

pS ( 1 − pS ) n

Example: A random sample of 400 voters showed 32 preferred candidate A. Set up a 95% confidence interval estimate for p. Ps = 32/400 = .08

ps − Z α / 2 ⋅ .08 − 196 . ⋅

ps ⋅ (1 − ps ) ps ⋅ (1 − ps ) ≤ p ≤ ps + Z α / 2 ⋅ n n .08 ⋅ (1−.08 ) 400

. ⋅ ≤ p ≤ .08 + 196

.053 ≤ p ≤ .107

.08 ⋅ (1−.08) 400