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 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. 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. 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.

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To investigate the proportion of new cars that had a mechanical problem within the first 5,000 miles driven, the automobile manufacturer used a simple random sample of 1,000 customers from a list of 30,000 customers who bought the new cars. A simple random sample is a subset of a population that is selected randomly, where each member of the population has an equal chance of being selected. This method ensures that the sample fairly represents the characteristics of the population under study. Each customer from the sample 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. However, 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. This is because 0.325 is based on a small and conditional sample of only 40 customers who reported any problem, not on the whole sample of 1,000 customers or the whole population of 30,000 customers. Therefore, it is not representative of the population parameter and may have a large sampling error. A better 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 13/1000 x 30,000 = 390. This is based on the ratio of customers who reported a problem with the power door locks to the whole sample size, multiplied by the population size.