Statistical Inference: Confidence Intervals & Hypothesis Testing for Means & Proportions, Study notes of Statistics

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STAT 301 TA : Lisa Chung [email protected]

DISCUSSION 11

(Mar. 11. 2004)

Two Samples

Means, when independent

• (1 − α)100% Confidence Interval for μ 1 − μ 2

(( ¯x 1 − x¯ 2 ) − z α 2

√

s^21 n 1

+

s^22 n 2 , ( ¯x 1 − x¯ 2 ) + z α 2

√

s^21 n 1

+

s^22 n 2

)

• Test statistic for H 0 : μ 1 − μ 2 = μ 0

Z =

( X¯ 1 − X¯ 2 ) − μ 0 √ s^21 n 1 +^

s^22 n 2

N (0, 1)

When two samples are dependent, calculate the difference of each matched pair of observatiosthereby forming a single collapsed sample, then apply the appropriate one sample test.

Small samples, when independent

• If n 1 < 30 and/or n 2 < 30, then use t-distribution, provided H 0 : σ^21 = σ^22 , informally, 14 < s

(^21) s^22 <^ 4.

• Then common value of σ^21 and σ^22 can be estimated by the weighted mean of s^21 and s^22

s^2 pooled =

(n 1 − 1)s^21 + (n 2 − 1)s^22 n 1 + n 2 − 2

s.e.ˆ =

√

s^2 pooled(

n 1

+

n 2

)

• (1 − α)100% Confidence Interval for μ 1 − μ 2

(( ¯x 1 − x¯ 2 ) − tdf, α 2

√

s^2 pooled(

n 1

+

n 2

), ( ¯x 1 − x¯ 2 ) + tdf, α 2

√

s^2 pooled(

n 1

+

n 2

))

, df= n 1 + n 2 − 2

• Test statistic for H 0 : μ 1 − μ 2 = μ 0 T = ( ¯X^1 −^ q X¯^2 )−μ^0 s^2 pooled( (^) n^11 + (^) n^12 ) tdf

Variance If X 1 N (μ 1 , σ 1 ),X 2 N (μ 2 , σ 2 ), H 0 : σ^21 = σ^22 , then, F = s

(^21) s^22 Fn^1 −^1 ,n^2 −^1

Proportion

Office: 1335 MSC, 263-5948 1 Office Hour: Wed.1:00-2:00 and Thurs. 11:00-12:

STAT 301 TA : Lisa Chung [email protected]

• The confidence interval for π 1 − π 2

(( ˆπ 1 − πˆ 2 ) − zα/ 2

√

π ˆ 1 (1 − πˆ 1 ) n 1

+

πˆ 2 (1 − πˆ 2 ) n 2 , ( ˆπ 1 − πˆ 2 ) + zα/ 2

√

π ˆ 1 (1 − πˆ 1 ) n 1

+

πˆ 2 (1 − πˆ 2 ) n 2

)

• Test for H 0 : π 1 − π 2 = 0 πpooledˆ = X n^11 ++Xn 22 , s.e. 0 =

√

πpooled(1 − πpooled) ∗

√

1 n 1 +^

1 n 2 Test statistic Z = ( ˆπ^1 − s.e.^ πˆ^20 ) −^0 N(0,1)

• Alternative method- χ^2 test for H 0 : π 1 − π 2 = 0 X^2 =

∑ (^) (Obs−Exp) 2 Exp χ 2 1

Example 1. The arrival time of my usual morning bus,B, is normally distributed with a neam ETA at 8 a.m. and a s.d. of 4 minutes. My arrival time, A, at the bus stop is also normally distributed with a mean ETA at 7 : 50 a.m.and a s.d. of 3 minutes. With what probability can I expect to catch the bus?

Example 2.Assume that population cholesterol level is normally distributed. a. Consider a small clinical trial, design to measure the efficacy of a new cholesterol-lowing drug against placebo. A group of six high-cholesterol patients is randomized to either a treatment arm of a control arm. b. Now imagine that the same drug is tested using another pilot study, with a different design. Serum cholesterol levels of 3 patients are measured at the beginning of the study, then remeasured after six month treatment period on the drug.

Example 3.Test of independence Imagine that a marketing research study surveys a random sample of n=2000 consumers about their responses regarding two brands of a certain product.Consider the null hypothesis, H 0 : πA|B = πA|Bc , i.e. the probability of liking A, given that B is liked , is equal to probability of liking A, givan that B is not liked.

Office: 1335 MSC, 263-5948 2 Office Hour: Wed.1:00-2:00 and Thurs. 11:00-12: