Tutorial Solution Week9
BES Tutorial Sample Solutions, S1/13
WEEK 9 TUTORIAL EXERCISES (To be discussed in the week starting May 6) 1. Perform the following hypothesis tests of the population mean. In each case, illustrate the rejection regions on both the Z and ̅ distributions, and calculate the p-value (prob-value) of the test. (a) H0: μ = 50, H1: μ > 50, n = 100, ̅ = 55, σ = 10, α = 0.05 Rejection region: ̅ 50 1.645 . 10⁄√100 Alternatively 10 51.645 50 1.645 ̅ ̅ . √ √100 Since 55 50 5 1.645 . 10⁄√100 Can reject H 0 and conclude that the population mean is greater than 50 .
0.05
50
51.645 reject
X
1
0.05 0 1.645 reject Z
5 (b)
0.0000
H0: μ = 25, H1: μ < 25, n = 100, ̅ = 24, σ = 5, α = 0.1 ̅ 25
.
Rejection region:
Alternatively ̅ Since ̅
5⁄√100
1.28
.
√
25
1.28
5 √100
24.36
24
25
Can reject H 0 and conclude that the population mean is less than 25 . 0.1 X 24.36 25 reject
2
5⁄√100
2
.
1.28
0.1 ‐1.28 0 Z
reject 2 0.0228
(c) H0: μ = 80, H1: μ ≠ 80, n = 100, ̅ = 80.5, σ = 4, α = 0.05 Rejection region: ̅
80
.
Alternatively: ̅ or ̅ Since ̅ ̅
4⁄√100
1.96
1.96
.
√ √ 80.5
80 80
1.96 1.96
4 √100 4 √100
79.216 80.784
.
80
is not less than ‐1.96 or nor greater than 1.96 we do not reject H 0 and conclude that the population mean is equal to 80.
3
4⁄√100
1.25
0.025
0.025
79.216 reject
80
80.784
X reject 0.025
0
1.25
0.025
‐1.96 reject 2
1.96 2
Z
reject 0.1056 0.2112
4
2. A real estate expert claims the current mean value of houses in a particular area is more than $250,000. A random sample of 150 recent sales prices in
WEEK 9 TUTORIAL EXERCISES (To be discussed in the week starting May 6) 1. Perform the following hypothesis tests of the population mean. In each case, illustrate the rejection regions on both the Z and ̅ distributions, and calculate the p-value (prob-value) of the test. (a) H0: μ = 50, H1: μ > 50, n = 100, ̅ = 55, σ = 10, α = 0.05 Rejection region: ̅ 50 1.645 . 10⁄√100 Alternatively 10 51.645 50 1.645 ̅ ̅ . √ √100 Since 55 50 5 1.645 . 10⁄√100 Can reject H 0 and conclude that the population mean is greater than 50 .
0.05
50
51.645 reject
X
1
0.05 0 1.645 reject Z
5 (b)
0.0000
H0: μ = 25, H1: μ < 25, n = 100, ̅ = 24, σ = 5, α = 0.1 ̅ 25
.
Rejection region:
Alternatively ̅ Since ̅
5⁄√100
1.28
.
√
25
1.28
5 √100
24.36
24
25
Can reject H 0 and conclude that the population mean is less than 25 . 0.1 X 24.36 25 reject
2
5⁄√100
2
.
1.28
0.1 ‐1.28 0 Z
reject 2 0.0228
(c) H0: μ = 80, H1: μ ≠ 80, n = 100, ̅ = 80.5, σ = 4, α = 0.05 Rejection region: ̅
80
.
Alternatively: ̅ or ̅ Since ̅ ̅
4⁄√100
1.96
1.96
.
√ √ 80.5
80 80
1.96 1.96
4 √100 4 √100
79.216 80.784
.
80
is not less than ‐1.96 or nor greater than 1.96 we do not reject H 0 and conclude that the population mean is equal to 80.
3
4⁄√100
1.25
0.025
0.025
79.216 reject
80
80.784
X reject 0.025
0
1.25
0.025
‐1.96 reject 2
1.96 2
Z
reject 0.1056 0.2112
4
2. A real estate expert claims the current mean value of houses in a particular area is more than $250,000. A random sample of 150 recent sales prices in