Biostatistics question

Question

3. Suppose a researcher is interested in the random variable detailing average distance for patients traveled to the nearest health center decides to carry out a Z test. He collects a simple random sample of 100 patients sampled from a population with a known standard deviation of 32.8 miles.
A) If the researcher wants to be 95% confident about his estimate, calculate the margin of error based on the above information.


B) Carry out the same calculation above, but assuming the researcher wishes to be “virtually certain” about his estimate. (Use Z=3.0)

Answer 1

To calculate the margin of error for a 95% confidence interval, you can use the standard formula for the margin of error in a Z-test:

Margin of Error (ME) = Z * (σ / √n)

Where:

1. Z is the Z-score corresponding to the desired confidence level (for a 95% confidence level, Z is approximately 1.96).
2. σ is the population standard deviation (32.8 miles in this case).
3. n is the sample size (100 patients).

Substituting the values:

ME = 1.96 * (32.8 / √100) = 1.96 * (32.8 / 10) = 6.4288 miles

So, the margin of error for a 95% confidence interval is approximately 6.43 miles.

B) If the researcher wishes to be "virtually certain" about his estimate and uses a Z-score of 3.0, you can use the same formula with Z = 3.0:

ME = 3.0 * (32.8 / √100) = 3.0 * (32.8 / 10) = 9.84 miles

So, with a Z-score of 3.0, the margin of error for the confidence interval is approximately 9.84 miles. This wider margin provides a higher level of confidence in the estimate.