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QuestionStatistics
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. 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. 400 customers from the sample reported a problem. Of the 400 customers who reported a problem, 140 customers, or 35%, reported a problem specifically with the power door locks.
Explain why 0.35 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.
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Answer
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Step 1:Let me solve this step by step with precise statistical reasoning:
Step 2:: Understand the Sampling Context
The manufacturer surveyed a sample of 1,000 customers out of 30,000 total car buyers. In this sample, 400 customers reported mechanical problems, and 140 of those specifically reported power door lock issues.
Step 3:: Identify the Sampling Bias
The proportion $$\frac{140}{400} = 0.35$$ represents only the proportion of problems within the subset of customers who already reported mechanical issues, NOT the proportion of all car buyers.
Step 4:: Explain the Statistical Error
Using 0.35 would be incorrect because: - This percentage is conditional on ALREADY having a mechanical problem - It does not represent the overall proportion of cars with power door lock issues - This subset represents only customers who experienced mechanical problems, which is a biased sample
Step 5:: Demonstrate the Correct Calculation
\frac{140}{1,000} = 0.14$$ or 14% of the total sample
The correct approach would be to calculate the proportion based on the entire sample:
Final Answer
0.35 should not be used because it represents a conditional proportion within a subset of problematic cars, not the true population proportion of cars with power door lock issues across all 30,000 cars sold.
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