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

To develop a sampling plan for obtaining a simple random sample of 1,000 customers from the list of 30,000 who purchased new cars, we can follow these structured steps:

1. Define the Target Population - The target population consists of all customers who bought a new car from the manufacturer in the specified year, totaling 30,000 individuals.

2. Determine the Sample Size - We aim to select a sample size of 1,000 customers from the total population of 30,000.

3. Create a Sampling Frame - Utilize the available list of names and addresses of all customers to ensure that every individual has an equal chance of being selected.

4. Random Selection - Employ a random number generator or a similar method to select 1,000 unique customers from the list. This ensures that each customer has an equal probability of being included in the sample, thus maintaining the integrity of a simple random sample.

5. Data Collection - Once the sample is selected, contact the customers through mail or digital means to administer a questionnaire regarding any mechanical problems experienced within the first 5,000 miles.

For part (b), the reason why the proportion of 0.325 should not be used to estimate the population proportion of new cars that experienced problems with the power door locks is that this percentage is derived only from the subset of customers who reported mechanical issues. Specifically, it reflects the proportion of those who had problems among the 40 customers who reported any issues, rather than the entire population of 30,000. This statistic fails to account for the majority of customers who did not experience any problems, leading to a biased estimation.

For part (c), to estimate the number of new cars sold that experienced a problem with the power door locks within the first 5,000 miles, we apply the point estimate derived from the sample. Given that 32.5% of the 40 customers who reported problems had issues with the power door locks, we can calculate the expected number of cars with this specific problem in the total population. The calculation is as follows:

Point Estimate = Total Cars Sold * Proportion with Power Door Lock Issues

Point Estimate = 30,000 * 0.325 = 9,750

However, since the question specifies that the estimate is based on the sample of 1,000 customers, we find that 390 new cars are estimated to have experienced a problem with the power door locks within the first 5,000 miles driven.