Download Lecture 10 Confidence Intervals Confidence

Lecture 10 Confidence Intervals
Confidence Interval for One Population Mean
Given an independent and identically distributed (iid) sample of size n, the C % confidence interval
for µ is given by
x̄ ± q ∗ SEx̄ ,
where x̄ is the sample mean, SEx̄ is the standard error of the sample mean, and q ∗ is the appropriate
quantile from the Z or t distribution.
• If the population variance σ 2 is known:
σ
x̄ ± z(1−C%)/2 × √ .
n
where z(1−C%)/2 is the (1 − C%)/2 quantile of the standard Normal distribution.
• If the population variance σ 2 is unknown:
s
x̄ ± t(1−C%)/2, n−1 × √ .
n
where t(1−C%)/2, n−1 is the (1 − C%)/2 quantile of the t distribution with degrees of freedom
n − 1 and s is the sample standard deviation.
Confidence Interval for the Difference in Two Population Means
Given two iid samples of size n1 and n2 from two populations, the C % confidence interval for the
difference in means µ1 − µ2 is
(x1 − x2 ) ± q ∗ SEx1 −x2 .
• If the population variances σ12 and σ22 are known:
s
(x1 − x2 ) ± z(1−C%)/2 ×
σ12 σ22
+ .
n1 n2
• If the population variances σ12 and σ22 are unknown and equal:
r
1
1
(x1 − x2 ) ± t(1−C%)/2, n1 +n2 −2 × sp
+ .
n1 n2
where sp is the pooled variance
s
sp =
(n1 − 1)s21 + (n2 − 1)s22
.
n1 + n 2 − 2
1
• If the population variances σ12 and σ22 are unknown and unequal:
s
s2
s21
+ 2
(x1 − x2 ) ± t(1−C%)/2, tWS ×
n1 n2
where tW S is the Welch-Satterthwaite equation
tWS =
(s21 /n1 + s22 /n2 )2
.
(s21 /n1 )2 /(n1 − 1) + (s22 /n2 )2 /(n2 − 1)
• If the two populations are paired. Define the difference between each pair of observations
D = X1 − X2 , then
sD
D̄ ± t(1−C%)/2, n−1 × √ .
n
Example: STA103 Student Survey Summary Statistics
Variable
Height (in.)
Homework (hr)
Reading (hr)
Homework-Reading (hr)
TV per day (hr)
Overall (n = 72)
Mean
S.D
67.29
5.60
3.41
1.52
2.78
1.81
0.63
1.82
0.74
1.43
Male (n = 46)
Mean
S.D
69.85
5.80
3.24
1.58
2.75
1.99
0.45
1.99
0.63
0.78
Find the 90% confidence interval for:
1. Height for the class.
2. Difference in height between male and female students.
3. Average number of hours per day spent on watching TV.
4. Hours per week for homework.
5. Difference between hours spent on reading and on homework.
2
Female (n = 26)
Mean
S.D
64.50
3.00
3.72
1.36
2.84
1.46
0.96
1.45
0.94
2.16