Question

1) A financial analyst would like to construct a 95% confidence interval for the mean earnings of a company. The company's earnings have a standard deviation of $12 million. What is the minimum sample size required by the analyst if he wants to restrict the margin of error to $2 million?

2) A budget airline wants to estimate what proportion of customers would pay $10 for in-flight wireless access. Given that the airline has no prior knowledge of the proportion, how many customers would it have to sample to ensure a margin of error of no more than 5% for a 95% confidence interval?

3) A job candidate with an offer from a prominent investment bank wanted to estimate how many hours she would have to work per week during her first year at the bank. She took a sample of six first-year analysts, asking how many hours they worked in the last week. Construct and interpret a 95% confidence interval with her results: 64, 82, 74, 73, 78, and 87 hours.

Answer 1

1)
The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 2, σ = 12

The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.96 * 12/2)^2
n = 138.3

Therefore, the sample size needed to satisfy the condition n >= 138.3 and it must be an integer number, we conclude that the minimum required sample size is n = 139
Ans : Sample size, n = 139 or 138

2)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.05

The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.5*(1 - 0.5)*(1.96/0.05)^2
n = 384.16

Therefore, the sample size needed to satisfy the condition n >= 384.16 and it must be an integer number, we conclude that the minimum required sample size is n = 385
Ans : Sample size, n = 385 or 384

3)
sample mean, xbar = 76.3333
sample standard deviation, s = 7.9666
sample size, n = 6
degrees of freedom, df = n - 1 = 5

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.571

ME = tc * s/sqrt(n)
ME = 2.571 * 7.9666/sqrt(6)
ME = 8.362

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (76.3333 - 2.571 * 7.9666/sqrt(6) , 76.3333 + 2.571 * 7.9666/sqrt(6))
CI = (68.97 , 84.70)