Problem
Score: $3.6 / 10 \quad 5 / 10$ answered Question 6 You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $\sigma=32.3$. You would like to be $99 \%$ confident that your esimate is within 0.2 of the true population mean. How large of a sample size is required? \[ n= \] Do not round mid-calculation. However, use a critical value accurate to three decimal places - this is important for the system to be able to give hints for incorrect answers. Question Help: $\square$ Message instructor Check Answer
Solution
Step 1 ：The formula for the sample size in this case is given by: \(n = \left(\frac{Z_{\alpha/2} \cdot \sigma}{E}\right)^2\) where: \(Z_{\alpha/2}\) is the z-score corresponding to the desired confidence level (in this case, 99% confidence corresponds to a z-score of 2.576), \(\sigma\) is the population standard deviation (32.3 in this case), \(E\) is the desired margin of error (0.2 in this case).
Step 2 ：We can substitute the given values into the formula to find the required sample size.
Step 3 ：The `math.ceil` function is used to round up to the nearest whole number, since we can't have a fraction of a sample.
Step 4 ：Final Answer: The required sample size is \(\boxed{173076}\).
