Math 220 : Elementary Statistics Kane Nashimoto
Exam 3 Solutions
Spring 2007
1. Inference about mean with σunknown.
(a) H0:µ= 1.10 vs. Ha:µ6= 1.10
t∗=¯x−µ0
s/
√n=1.16 −1.10
0.17/
√20 = 1.578
Critical values : ±2.861 (t.01/2,20−1= 2.861)
Retain H0. (µ≈1.10)
(b) ¯x±t.01/2,20−1
s
√n; 1.16 ±2.861 0.17
√20 ; 1.16 ±0.109 ; (1.051 ,1.269)
(c) The 99% confidence interval should contain µ0= 1.10. The result of (a) indicates
that 1.10 is a plausible value of µand, therefore, must be contained in the
confidence interval.
2. Inference about proportion.
(a) p= 273/390 = .700
p±z.05/2rp(1 −p)
n;.700 ±1.960r(.700)(1 −.700)
390 ;.700 ±.045 ; (.655 , .745)
(b) Let π=p.
n=π(1 −π)z.05/2
B2
= (.700)(1 −.700)1.960
.04 2
= 504.21 ⇒505
3. Sampling distribution.
(a) µ¯
X=µ= 61.0
(b) σ¯
X=σ/
√n= 4.0/
√20
(c) P(¯
X > 60.0) = P¯
X−µ
σ/
√n>60.0−61.0
4.0/
√20 =P(Z > −1.12)
= 1 −P(Z≤ −1.12) = 1 −.1314 = .8686
4. Inference about mean with σknown.
(a) H0:µ= 33.0 vs. Ha:µ > 33.0
z∗=¯x−µ0
σ/
√n=35.0−33.0
4.0/
√30 = 2.739
Critical value : +1.645 (z.05 = 1.645)
Reject H0. (µ > 33.0)
(b) p-value = P(Z≥2.75) = 1 −P(Z < 2.75) = 1 −.9970 = .0030
