Answer :
Final answer:
To obtain a sample mean that estimates the true mean within a 1-point margin of error at a 99.7% confidence level, the required standard deviation for the sample mean is determined using the population standard deviation and the z-score corresponding to the desired confidence level.
Explanation:
To estimate the mean score μ of those who took the Medical College Admissions Test with an error of no more than 1 point in either direction at the 99.7% confidence level, we need to determine the standard deviation of the sample mean x that satisfies this condition.
Steps to Determine the Required Standard Deviation
Identify the z-score for the desired confidence level. Since 99.7% confidence corresponds to a z-score of 3 (as per the empirical rule), this is used for calculations.
Use the equation for the standard error (SE) of the sample mean, which is the population standard deviation (σ) divided by the square root of the sample size (n): SE = σ/√n.
To find the maximum standard deviation that x can have, set the z-score (z) times the SE to be less than or equal to the maximum error allowed (E): z × SE ≤ E. In this case, 3 × SE ≤ 1.
Since the population standard deviation is given as 6.4, plug this value into the previous step to solve for n. SE = 6.4/√n and we want 3 × (6.4/√n) ≤ 1, so √n ≥ (3 × 6.4)/1.
Solve for n and then calculate the required standard deviation for the sample mean, which will be σ/√n.
By following these steps, we ensure that 99.7% of all samples give a sample mean x within 1 point of μ, adhering to Central Limit Theorem and statistical confidence interval principles.
