You Have Just Been Hired By General Motors To Tour The Unite

You Have Just Been Hired By General Motors To Tour The United States

You have just been hired by General Motors to tour the United States giving randomly selected drivers test rides in a new Corvette (yeah, right!). After giving the test drive, you must ask the rider whether he or she would consider buying a Corvette. How many riders must you survey (take for test rides) to be 90% confident that the sample proportion is off by no more than five percentage points?

Paper For Above instruction

In conducting market research to estimate the proportion of potential buyers interested in purchasing a new Corvette, it is essential to determine an adequate sample size that balances confidence level, margin of error, and variance. The goal is to find out how many test rides must be conducted to ensure that the estimated proportion is within a specified margin of error with a specified level of confidence.

The problem specifies a confidence level of 90% and a margin of error (also known as the maximum allowable difference between the sample proportion and the true population proportion) of five percentage points, or 0.05. To determine the required sample size, we utilize the formula derived from the confidence interval estimation of a population proportion:

n = (Z^2  p  (1 - p)) / E^2

Where:

  • n is the required sample size
  • Z is the Z-score corresponding to the desired confidence level
  • p is the estimated proportion of interest (if unknown, use p = 0.5 to maximize the required sample size)
  • E is the margin of error

For a 90% confidence level, the Z-score (Z_{0.05}) is approximately 1.645. Since the true proportion is unknown, the conservative approach is to assume p = 0.5, which yields the largest possible sample size needed.

Plugging in the values:

n = (1.645^2  0.5  0.5) / 0.05^2

Calculating step by step:

  • 1.645 squared is approximately 2.706
  • 0.5 * 0.5 = 0.25
  • 0.05 squared is 0.0025

Thus,

n = (2.706 * 0.25) / 0.0025 = 0.6765 / 0.0025 = 270.6

Since sample size must be a whole number, we round up to the next integer: 271.

Therefore, to be 90% confident that the sample proportion of potential Corvette buyers is within five percentage points of the true proportion, you must survey at least 271 drivers.

This conservative estimate ensures adequate statistical power regardless of the actual proportion interested in purchasing a Corvette. Increasing the sample size beyond this minimum can further reduce the margin of error or increase confidence but is not necessary according to the current parameters.

References

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