An automobile manufacturer sold 30,000 new cars, one to each of 30,000 customers, in a certain year. The manufacturer was intere

Question

2 months ago

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An automobile manufacturer sold 30,000 new cars, one to each of 30,000 customers, in a certain year. The manufacturer was intere

sted in investigating the proportion of the new cars that experienced a mechanical problem within the first 5,000 miles driven. (a)
A list of the names and addresses of all customers who bought the new cars is available. Describe a sampling plan that could be used to obtain a simple random sample of 1,000 customers from the list.

Each customer from a simple random sample of 1,000 customers who bought one of the new cars was asked whether they experienced any mechanical problems within the first 5,000 miles driven. Forty customers from the sample reported a problem. Of the 40 customers who reported a problem, 13 customers, or 32.5%, reported a problem specifically with the power door locks. (b)

Explain why 0.325 should not be used to estimate the population proportion of the 30,000 new cars sold that experienced a problem with the power door locks within the first 5,000 miles driven. (c)

Based on the results of the sample, give a point estimate of the number of new cars sold that experienced a problem with the power door locks within the first 5,000 miles driven.

Mathematics

1 answer:

babunello [11.8K] · Answer 1 · 2026-02-16T09:03:00+03:00

A) To establish a sampling plan, follow these 5 steps:
1) Identify the sample population: which customers will you reach out to?
   Those who purchased a new car during a specific year.
2) Determine the population size: how many customers will you contact?
   From the 30,000 car buyers, select 1,000 customers to contact.
3) Select contact method: what is your means of contacting customers?
   Since you have a list with names and addresses, mailing questionnaires is feasible.
4) Define the sampling frame: what is the timeline or deadline for contacting customers?
   Send out questionnaires and allow two months for responses.
5) Decide on the analysis approach: is your research qualitative or quantitative?
   You aim for quantitative research, so you will use probabilistic sampling.

B) The 32.5% probability relates only to those customers who experienced mechanical issues, specifically power door lock problems, and does not account for those without any problems or those reporting issues after the first 5,000 miles.

C) To estimate the probability of power door lock problems among all customers within the first 5,000 miles, consider the entire sample:
P = 13 / 1000 = 0.013

Therefore,
N = 0.013 × 30,000 = 390

Thus, the estimated number of new cars that had power door lock issues within 5,000 miles is 390.