Use the given information to find the minimum sample size required to estimate an unknown population mean μ.
How many commuters must be randomly selected to estimate the mean driving time of Chicago commuters? We want 99% confidence that the sample mean is within 4 minutes of the population mean, and the population standard deviation is known to be 6 minutes.
I have no idea how to go about solving the above question. Please help if you can.
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To find the minimum sample size required to estimate an unknown population mean μ, we can use the formula:
n = (Z * σ / E)²
where:
n is the sample size
Z is the z-score corresponding to the desired confidence level (in this case, we want 99% confidence)
σ is the population standard deviation
E is the maximum error or margin of error desired
In this case, we want 99% confidence that the sample mean is within 4 minutes of the population mean, and the population standard deviation is known to be 6 minutes. So we can plug in the values into the formula:
n = (Z * σ / E)²
First, we need to find the value of Z for a 99% confidence level. The z-score can be found using a standard normal distribution table or a statistical software. For a 99% confidence level, the corresponding z-score is approximately 2.576.
n = (2.576 * 6 / 4)²
Simplifying the equation:
n = (15.456 / 4)²
n = 3.864²
n ≈ 14.9376
Rounding up to the next whole number, the minimum sample size required is approximately 15 commuters.
Therefore, you would need to randomly select at least 15 commuters to estimate the mean driving time of Chicago commuters with 99% confidence that the sample mean is within 4 minutes of the population mean, assuming a known population standard deviation of 6 minutes.