Question

1. 33 packages are randomly selected from packages received by a parcel service. The sample has a mean weight of 14.3 pounds. Assume the population standard deviation of all such packages is 3.9 pounds. What is the 97% confidence interval for the true mean weight, μ, of all packages received by this parcel service?

2. A Statistics student wants to estimate the average delivery time for Uber Eats with 99% confidence. If they assume the population standard deviation of delivery times is 10 minutes with a margin of error of 2.5 minutes, how large must the sample be?

3.In a survey of 1000 people, 700 said that they voted in a recent presidential election. Voting records show that 61% of eligible voters actually did vote. Construct a 96% confidence interval estimate of the proportion of people who say that voted.

Answer 1

1)

sample mean, xbar = 14.3
sample standard deviation, σ = 3.9
sample size, n = 33

Given CI level is 97%, hence α = 1 - 0.97 = 0.03
α/2 = 0.03/2 = 0.015, Zc = Z(α/2) = 2.17

ME = zc * σ/sqrt(n)
ME = 2.17 * 3.9/sqrt(33)
ME = 1.47

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (14.3 - 2.17 * 3.9/sqrt(33) , 14.3 + 2.17 * 3.9/sqrt(33))
CI = (12.83 , 15.77)

2)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 2.5, σ = 10

The critical value for significance level, α = 0.01 is 2.58.

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 = (2.58 * 10/2.5)^2
n = 106.5

Therefore, the sample size needed to satisfy the condition n >= 106.5 and it must be an integer number, we conclude that the minimum required sample size is n = 107
Ans : Sample size, n = 107 or 106

3)

sample proportion, = 0.7
sample size, n = 1000
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.7 * (1 - 0.7)/1000) = 0.014

Given CI level is 96%, hence α = 1 - 0.96 = 0.04
α/2 = 0.04/2 = 0.02, Zc = Z(α/2) = 2.05

Margin of Error, ME = zc * SE
ME = 2.05 * 0.014
ME = 0.0287

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.7 - 2.05 * 0.014 , 0.7 + 2.05 * 0.014)
CI = (0.6713 , 0.7287)