Question

iPhonesSuppose $1000$iPhones are produced at a factory today. Management would like to ensure that the phones’ display screens meet their quality standards before shipping them to retail stores. Because it takes about $10$minutes to inspect an individual phone’s display screen, managers decide to inspect a sample of $20$phones from the day’s production.

a. Explain why it would be difficult for managers to inspect an SRS of $20$iPhones that are produced today.

b. An eager employee suggests that it would be easy to inspect the last $20$iPhones that were produced today. Why isn’t this a good idea?

c. Another employee recommends a different sampling method: Randomly choose one of the first$50$ iPhones produced. Inspect that phone and every fiftieth iPhone produced afterward. (This method is known as systematic random sampling.) Explain carefully why this sampling method is not an SRS.

Short answer

a. If a random sample of size $20$is taken from $1000$iPhones produced, the sample size $1000$is quite small compared to the overall output of the day, with $0.001$chance of being chosen.

b. If only the final $20$phones are examined, this sample will not be typical of the entire output., at the end of the day, there may be a change in the production process (maybe the machine isn't performing efficiently.

c. SRS is a type of probability sampling in which every member of the population has an equal chance of being selected. Systematic sampling is a hybrid of probabilistic and non-probabilistic sampling techniques.

Step-by-step solution

Part (a) Step 1 : Given Information

We have to explain why it would be difficult for managers to inspect an SRS of $20$iPhones that are produced today.

Part (a) Step 2 : Simplification

If a random sample of size $20$is taken from $1000$iPhones produced, the sample size $20$is quite small compared to the overall output of the day.

Each of the $20$units has a probability in an SRS of $20$units out of $1000$units.

$\frac{1}{1000}=0.001$chance of being chosen, which is extremely low.

Part (b) Step 1 : Given Information

We have to explain why the idea given in the question is not good.

Part (b) Step 2 : Simplification

If only the final $20$phones are examined, this sample will not be typical of the entire output.

Because, at the end of the day, there may be a change in the production process (maybe the machine isn't performing efficiently.

Any change in the manufacturing process has an impact on the quality specifications of a fixed number of iPhones manufactured in the future. The management wants to make sure that all $1000$iPhones produced fulfil the display screen quality specifications.

As a result, evaluating the last $20$phones produced does not guarantee that the entire day's production will fulfil the quality standard.

Part (c) Step 1 : Given Information

We have to explain why this sampling method is not an SRS.

Part (c) Step 2 : Simplification

SRS is a type of probability sampling in which every member of the population has an equal chance of being
selected.

Systematic sampling is a hybrid of probabilistic and non-probabilistic sampling techniques.

Because the initial member of the sample is chosen at random from the first $50$units, this sampling method is probabilistic (each of first $50$population units has Equal chance of being selected as first sample unit).

Theremainingunitsinthesamplearefixed(everyfiftiethmemberisselectedafterthefirstunitisselected)bythechoiceofthefirstunit,makingthismethodnon-probabilistic.

As a result, systematic random sampling does not qualify as an SRS.