A local package delivery service wants to improve the efficiency of its deliveries. As a first step, the management team decides to conduct a study to determine the average length of time from the arrival of a package at the company's mail center until its delivery at a home. Which of the following methods is most likely to produce valid results? $\newline$ Choose $1$ answer: $\newline$ (A) The team selects the $1,000$ heaviest packages in a one-week, non-holiday period and records how long it takes for each package to reach its destination. $\newline$ (B) The team calls $1,000$ residents in their delivery area and asks them whether they have received a package from their service in the past week. They will then record how long it took for those packages to reach their destination. $\newline$ (C) The team selects a random sample of $1,000$ packages arriving at the center over a one-week, non-holiday period and records how long it takes for each package to reach its destination. $\newline$ (D) None of the above

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Identify quotients of rational numbers: word problems

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Q. A local package delivery service wants to improve the efficiency of its deliveries. As a first step, the management team decides to conduct a study to determine the average length of time from the arrival of a package at the company's mail center until its delivery at a home. Which of the following methods is most likely to produce valid results?

\newline

Choose

1

answer:

\newline

(A) The team selects the

1,000

heaviest packages in a one-week, non-holiday period and records how long it takes for each package to reach its destination.

\newline

(B) The team calls

1,000

residents in their delivery area and asks them whether they have received a package from their service in the past week. They will then record how long it took for those packages to reach their destination.

\newline

1,000

packages arriving at the center over a one-week, non-holiday period and records how long it takes for each package to reach its destination.

\newline

(D) None of the above

Question Prompt: The question prompt for the new math problem is: ＂Which method is most likely to produce valid results for determining the average length of time from the arrival of a package at the company's mail center until its delivery at a home?＂
Evaluate Options: To solve this problem, we need to evaluate the options given and determine which one would provide the most valid and unbiased results for the study. We are not performing calculations in this step, but rather applying principles of statistical sampling.
Option (A): Option (A) suggests selecting the $1,000$ heaviest packages. This method would introduce bias because it does not represent the entire range of packages that the delivery service handles. The weight of a package could influence the delivery time, and therefore, this sample would not be representative of all packages.
Option (B): Option (B) involves calling $1,000$ residents and asking about their recent deliveries. This method could also introduce bias because it relies on the residents' memory and honesty. Additionally, it may not be a random sample of all deliveries, as not all residents may have received a package, and those who did might not accurately recall the delivery time.
Option (C): Option (C) proposes selecting a random sample of $1,000$ packages over a one-week, non-holiday period. This method is most likely to produce valid results because it does not discriminate based on package weight or rely on potentially faulty customer recollections. A random sample is more likely to be representative of the overall population of packages, which is essential for obtaining an accurate average delivery time.
Option (D): Option (D) states ＂None of the above,＂ which we can disregard because option (C) is a valid method for conducting the study.