Answer:
Design 1: 151.19≤x≤160.37
Design 2: 152.46≤x≤155.88
Explanation:
Confidence interval formula is expressed as;
CI = xbar ± Z×σ/√n
xbar is the mean of the sample
σ is the standard deviation
n is the sample size
z is the z score at 80% confidence level
For Design 1:
158.5 138.4 168.1 149.4 145.8 168.7 154.4 162.9
xbar = Sum of the samples/sample size
Sum of samples = 158.5+ 138.4+ 168.1+149.4+ 145.8+ 168.7+ 154.4+ 162.9
Sum of samples = 1246.2
Sample size = 8
xbar = 1246.2/8
xbar = 155.78
Standard deviation
σ = √\sum(x-xbar)²/N
σ = (158.5 - 155.775)² + ... + (162.9 - 155.775)²/8
= 821.075/8
= 102.634375
= √102.634375
= 10.13
σ = 10.13
CI = 155.78±(1.282×10.13/√8)
CI = 155.78±(1.282×3.5815)
CI = 155.78±(4.5915)
CI = {155.78-4.5915, 155.78+4.5915}
CI = {151.19, 160.37}
The range for the true mean is 151.19≤x≤160.37
For Design 2:
150.3 155.4 151.6 158.8 151.4 150.8 161.4 157.6 156.8 147.6
xbar = Sum of the samples/sample size
Sum of samples = 150.3+ 155.4+ 151.6+158.8 +151.4+ 150.8+ 161.4 +157.6+ 156.8 +147.6
Sum of samples = 1541.7
Sample size = 10
xbar = 1541.7/10
xbar = 154.17
Standard deviation
σ = (150.3 - 154.17)²+ ... + (147.6 - 154.17)²/10
σ = 177.681/10
σ = 17.7681
σ = √17.7681
σ = 4.22
CI = 154.17±(1.282×4.22/√10)
CI = 154.17±(1.282×1.3345)
CI = 154.17±(1.7108)
CI = {154.17-1.7108, 154.17+1.7108}
CI = {152.46, 155.88}
The range for the true mean is 152.46≤x≤155.88