Download Document

ENGR 610
Applied Statistics
Fall 2007 - Week 7
Marshall University
CITE
Jack Smith
Overview for Today

Review Hypothesis Testing, 9.1-9.3



Go over homework problem 9.2
Hypothesis Testing, 9.4-9.7





One-Sample Tests of the Mean
Testing for the Difference between Two Means
Testing for the Difference between Two Variances
Testing for Paired Data or Repeated Measures
Testing for the Difference among Proportions
Homework assignment
Hypothesis Testing

One-Sample Tests for the Mean






Z Test ( known)
t Test ( unknown)
Two-tailed and one-tailed tests
p-value
Connection with Confidence Interval
Z Test for the proportion
Null hypothesis





A “no difference” claim about a population parameter
under suspicion based on a sample
Tested by sample statistics and either rejected or
accepted based on critical test (Z, t, F, 2) value
Rejection implies that an alternative (the opposite)
hypothesis is more probable
Analogous to a mathematical ‘proof by contradiction’
or the legal notion of ‘innocent until proven guilty’
Only the null hypothesis involves an equality, while
the alternative hypothesis deals only with inequalities
Critical Regions

Critical value of test statistic (Z, t, F, 2,…)



Based on desired level of significance
Acceptance (null hypothesis) region, and a
Rejection (alternative hypothesis) region

One-tailed or two-tailed
Type I and Type II errors


Seek proper balance between Type I and II errors
Type I error - false negative


Null hypothesis rejected when in fact it is true
Occurs with probability 



 = level of significance - chosen!
(1- ) = confidence coefficient
Type II error - false positive


Null hypothesis accepted when in fact it is false
Occurs with probability 



 = consumer’s risk
(1- ) = power of test
Depends on , difference between hypothesized and actual
parameter value, and sample size
Z Test ( known) - Two-tailed

Critical value (Zc) based on chosen level of
significance, 



Typically  = 0.05 (95% confidence), where
Zc = 1.96 (area = 0.95/2 = 0.475)
 = 0.01 (99%) and 0.001 (99.9%) are also common, where
Zc = 2.57 and 3.29
Null hypothesis (<X> = µ) rejected if Z > Zc or < -Zc,
where
Z
X 

n
Z Test ( known) - One-tailed

Critical value (Zc) based on chosen level of
significance, 


Typically  = 0.05 (95% confidence), but where
Zc = 1.645 (area = 0.95 - 0.50 = 0.45)
Null hypothesis (<X> ≤ µ) rejected if Z > Zc, where
Z
X 

n
t Test ( unknown) - Two-tailed

Critical value (tc) based on chosen level of
significance, , and degrees of freedom, n-1


Typically  = 0.05 (95% confidence), where, for example
tc = 2.045 (upper area = 0.05/2 = 0.025), for n-1 = 29
Null hypothesis rejected if t > tc or < -tc, where
X 
t
s
n
t
Z Test on Proportion

Use normal approximation to binomial
distribution, where
p
Z
n (1  )

p-value vs critical value

Use probabilities corresponding to
values of test statistic (Z, t,…)





If the p-value  , accept null hypothesis
If the p-value < , reject null hypothesis
E.g., compare p to α instead of t to tc
More direct
Does not necessarily assume
distribution is normal
Connection with Confidence Interval



Compute the Confidence Interval for the
sample statistic (e.g., the mean) as in
Ch 8
If the hypothesized population
parameter is within the interval, accept
the null hypothesis, otherwise reject it
Equivalent to a two-tailed test

Double α for half-interval (one-tail) test
Z Test for the Difference
between Two Means


Random samples from independent groups
with normal distributions and known 1 and 2
Any linear combination (e.g., the difference) of
normal distributions (k, k) is also normal
   ak k
Z
k
 2   ak2 k2

 X2 
 12
n1
k
CLT:
( X 1  X 2 )  ( 1   2 )
 X2
n

 22
n2
'
Z
(X  X )
2
2
n
Populations 1 & 2 the same

t Test for the Difference between
Two Means (Equal Variances)

Random samples from independent groups
with normal distributions, but with equal and
unknown 1 and 2
(X 1  X 2 )  (1  2 )
H0: µ1 = µ2
tn1 n 2 2 
 1 1 
s   
n1 n2 
2
p

Using the pooled sample variance
2
2
(n
1)s

(n
1)s
1
2
2
s2p  1
(n1 1)  (n 2 1)
t Test for the Difference between
Two Means (Unequal Variances)

Random samples from independent groups
with normal distributions, with unequal and
unknown 1 and 2
tdf 



(X 1  X 2 )  (1  2 )
2
2
s1 s2

n1 n 2
Using the Satterthwaite approximation to the
degrees of freedom (df)
Use Excel Data Analysis tool!
F test for the Difference
between Two Variances

Based on F Distribution - a ratio of 2
distributions, assuming normal distributions
s12
F 2
s2


FL(,n1-1,n2-1)  F  FU(,n1-1,n2-1), where
FL(,n1-1,n2-1) = 1/FU(,n2-1,n1-1), and where
FU is given in Table A.7 (using nearest df)
Mean Test for Paired Data or
Repeated Measures


Based on a one-sample test of the
corresponding differences (Di)
Z Test for known population D
Z
D  D
D
H0: D = 0
n


t Test for unknown D (with df = n-1)
D  D
t
sD
n
2 Test for the Difference among
Two or More Proportions



Uses contingency table to compute
2
(
f

f
)
2   o e
Sum over all cells
fe
(fe)i = nip or ni(1-p) are the expected
frequencies, where p = X/n, and (fo)i are the
observed frequencies


For more than 1 factor, (fe)ij = nipj, where pj = Xj/n
Uses the upper-tail critical 2 value, with the
df = number of groups – 1

For more than 1 factor, df = (factors -1)*(groups-1)
Other Tests

2 Test for the Difference between Variances



Follows directly from the 2 confidence interval for
the variance (standard deviation) in Ch 8.
Very sensitive to non-Normal distributions, so not
a robust test.
Wilcoxon Rank Sum Test between Two
Medians
Homework



Work through rest of Appendix 9.1
Work and hand in Problems 9.69, 9.71,
9.74
Read Chapter 10

Design of Experiments: One Factor and
Randomized Block Experiments