Answer:
Step-by-step explanation:
Considering the central limit theorem, the distribution is normal since the number of samples is large.
Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
The sample mean, x is the point estimate for the population mean.
Confidence interval = mean ± z × σ/√n
Where
σ = population standard Deviation
σ/√n = sample standard deviation
Confidence interval = x ± z × σ/√n
1) x = $75
σ = $24
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.96 = 0.04
α/2 = 0.04/2 = 0.02
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.02 = 0.98
The z score corresponding to the area on the z table is 2.05. Thus, confidence level of 96% is 2.05
Confidence interval = 75 ± 2.05 × 24/√64
= 75 ± 2.05 × 3
= 75 ± 6.15
The lower end of the confidence interval is
75 - 6.15 = 68.85
The upper end of the confidence interval is
75 + 6.15 = 81.15
2) n = 400
x = $75
σ = $24
z = 2.05
Confidence interval = 75 ± 2.05 × 24/√400
= 75 ± 2.05 × 1.2
= 75 ± 2.46
The lower bound of the confidence interval is
75 - 2.46 = 72.54
The upper bound of the confidence interval is
75 + 2.46 = 77.46
3) n = 400
x = $200
σ = $80
The z score corresponding to the confidence level of 94% is 1.88
z = 1.88
Confidence interval = 200 ± 1.88 × 80/√400
= 200 ± 1.88 × 4
= 200 ± 7.52
Margin of error = 7.52
