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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 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.

b) 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.

c) 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.

d) 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.

Answer :

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 is [tex]\(0.325 \times 30,000 = 9,750\)[/tex] cars.

a) To obtain a simple random sample of 1,000 customers from the list of 30,000 customers who bought the new cars, the manufacturer can follow these steps:

1. Assign a unique identifier (such as a number) to each customer in the list.

2. Use a random number generator to select 1,000 unique identifiers from the list.

3. Select the customers corresponding to the selected identifiers to form the sample.

b) In the sample of 1,000 customers, 40 reported a mechanical problem within the first 5,000 miles driven. Among these 40 customers, 13 reported a problem specifically with the power door locks.

c) The proportion 0.325 (32.5%) 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 because:

1. The proportion is based on a sample of only 1,000 customers, which may not accurately represent the entire population of 30,000 new cars.

2. The sample may not be fully representative of the population, leading to potential sampling bias.

3. The sample size of 40 customers who reported a mechanical problem with the power door locks may be too small to provide a reliable estimate of the population proportion.

d) Based on the results of the sample, 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 is[tex]\(0.325 \times 30,000 = 9,750\) cars.[/tex]

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