The document discusses key concepts in hypothesis testing including: 1) A telescope manufacturer wants to test if a new telescope's standard deviation in resolution is below 2 when focusing on objects 500 light-years away based on a sample of 30 measurements with a standard deviation of 1.46. 2) Hypothesis testing involves a null hypothesis (H0) and alternative hypothesis (H1), and the two types of possible errors - Type I and Type II. 3) The probabilities of Type I and Type II errors depend on the critical region used to determine whether to reject the null hypothesis.

1. 8/13/2012
Confidence Interval of µ Student’s t-distribution
(σ unknown)
X −µ • Distribution of a cont. symmetric random variable
1 − α = P[−tα <
< tα ]
2
S
, ( n −1)
2
, ( n −1) with range (- ∞,∞). Similar appearance to Z except
n fat-tailed
S S • has a parameter, ν, known as degrees of
= P[ X − t α < µ < X + tα ]
2
, ( n −1) n 2
, ( n −1) n freedom(d.f.) N (0,1)
So, 100(1-α)% C.I. for µ is : S Standard → t with the d . f .of the denominator
X ± tα × χ2
error df
2
,( n −1) n
pt. estimate
table-value • If X1, X2,…,Xn are SRSWR from N(µ, σ2), then
(X − µ)
σ
Valid for sampling from a Normal population; n ( X − µ) n
= → t with ( n − 1) d . f .
if sample size n small) S (n − 1) S 2
If n is large, using z-value instead of t –value is ok 5
σ 2 ( n − 1)
6
Chi-square (χ2)Distribution Confidence Interval of σ2
• Distribution of a continuous (skewed) random variable with range (n − 1) S 2
(0,∞) 1 − α = P[ χ 2 α < 2
< χα ]
• has a parameter, ν, known as degrees of freedom(d.f.)
1− , n −1
2 σ2 2
, n −1
• mean = ν; variance = 2 ν (n − 1) S 2 (n − 1) S 2
• If X1, X2,…,Xn are SRSWR from N(0, 1), then = P[ 2
<σ 2 < ]
χα χ2 α
2 2 2 2 , n −1 1− , n −1
X + X + ⋯ + X → χ with n d . f .
1 2 n 2 2
2 2
A 90% C.I. (2-sided) of σ2 ( 29×1557 , 29×.1708 )
42.
.46
17
.46
• If X1, X2,…,Xn are SRSWR from N(µ, σ2), then
n
(n − 1) S 2 ∑(X i − X )2 Valid for sampling from a Normal population if n is small.
= i =1
→ χ 2 with (n − 1)d . f .
σ 2
σ 2
Example: LR 7.49* suppose we want C.I. for σ2
• Ex.: check that S2 is unbiased for σ2 and find S.E. of S2. LR Section 11.5. (problem Sc11-7, Problems 11-39 to11-41, 11-45
(modify others)
7 8
Aczel Sounderpandyan: Section 6.5
Statistical Hypothesis
Confidence Interval
Estimation and TOH
• Of two sample mean, proportion, variance Parameter
mean proportion Standard
σ 12
µ1 − µ 2 , π1 − π 2 , 2
deviation
σ2 Single
Population
Two
Multiple
9
1

2. 8/13/2012
Clear-Tone Radios Clear-Tone Radios:
H0: π ≤ 0.1 vs. H1: π > 0.1
Shankar had to decide whether the proportion of defective C.R. is X ≥c (c possibly 3,2,1,0)
items in a box exceeds 10% or not based on his Compute the possible Prob(type I error) for various c
inspection of 3 items randomly picked. c P ( T y p e I e r r o r )
3 .0 0 0 7
2
We have settled the problems as: .0 2 5 7
1 .2 7 3 4
Shankar should accept the lot if
Exercise: Compute the Prob[Type II error] for the different C.R.’s.
no more than 1 radio is defective out of 3.
Note that it depends on the true π (defective proportion). So, to get
an idea plot the values for certain π > 0.1
Power of a test = 1 - Probability[Type II error]
11 = P[rejecting H0 when H1 is true] 12
Problem Basic elements of TOH
The telescope manufacturer wants its telescopes to have standard • Hypothesis -- assertion about a parameter the truth of
deviations in resolutions to be significantly below 2 when focusing which is to be inferred based on sample/relevant statistic
on objects 500 light-years away. When a new telescope is used to • Null hypothesis (H0) vs. Alternative Hypothesis (H1)
focus on an object 500 light-years away 30 times, the (sample) • Two kinds of error (Type I error and Type II error)
standard deviation turns out to be 1.46.
D e c i s io n D o not R e je ct H 0
Should this telescope be sold? T ru th r eje c t H 0
H 0 is C o rrec t T ype I
co rrect d e c is i o n e rro r
H 1 is T yp e II C o rre ct
13 co rrect erro r d e c is i o n 14
• The two errors are NOT typically equi-important and this
decides which one should be the null hypothesis
• A test is usually specified by fixing the critical region (CR)
(you reject H0 if the sample falls here) or equivalently the
acceptance region which is its complement.
• Given a test (equivalently, a CR or the cutoff point), one
would like to compute the probabilities of Type I (and II) error
to gauge the consequences of wrong decisions
• With n fixed, P(type I error) increases with decrease in P(type
II error) and vice versa.
• The usual approach is
– Decide on acceptable level (called level of significance), α, for P(type I
error)
– determine the cutoff so that P(type I error) = α
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