Question

A local Chinese restaurant offers home delivery service for orders placed within 5Km of the restaurant. In order to determine whether or not it is providing good delivery service, the store manager wants to determine the average delivery time. From historical data, the standard deviation of delivery time is known to be 4.25 minutes.

a. After measuring the time taken for each of 50 home deliveries, the manager found that their average time was 12.45 minutes. Construct a 94% confidence interval for the average time taken to make home deliveries from this restaurant.

b. How many deliveries should be measured in this study in order to have the average delivery time estimated with a 45 second precision and 94% confidence? {45 second precision means you will be within ±45 seconds of the true mean}

c. What is the population for which inferencing was done in part (a)?

Answer 1

sample mean, xbar = 12.45
sample standard deviation, σ = 4.25
sample size, n = 50

Given CI level is 94%, hence α = 1 - 0.94 = 0.06
α/2 = 0.06/2 = 0.03, Zc = Z(α/2) = 1.88

ME = zc * σ/sqrt(n)
ME = 1.88 * 4.25/sqrt(50)
ME = 1.13

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (12.45 - 1.88 * 4.25/sqrt(50) , 12.45 + 1.88 * 4.25/sqrt(50))
CI = (11.32 , 13.58)

b)

The following information is provided,
Significance Level, α = 0.060 Margin or Error, E = 45, σ = 4.25*60 = 255

The critical value for significance level, α = 0.06 is 1.88.

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 = (1.88 * 255/45)^2
n = 113.49

Therefore, the sample size needed to satisfy the condition n >= 113.49 and it must be an integer number, we conclude that the minimum required sample size is n = 114
Ans : Sample size, n = 114

c)

time taken for deliveries from the restaurant