According to the distributive property, 3(a + b) = a)

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