Determine the minimum sample size required to be 99% confident that the sample mean driving distance to work is within 2 miles of the population mean driving distance to work. Use the population standard deviation from Exercise 2.

Verified step by step guidance

1. Identify the key components of the problem: the confidence level (99%), the margin of error (2 miles), and the population standard deviation (σ), which should be provided in Exercise 2. Denote the sample size as n.
2. Determine the z-score corresponding to a 99% confidence level. For a 99% confidence level, the z-score is the critical value that leaves 0.5% in each tail of the standard normal distribution. Use a z-table or statistical software to find this value.
3. Use the formula for the margin of error in estimating a population mean: \( E = z \cdot \frac{\sigma}{\sqrt{n}} \), where E is the margin of error, z is the z-score, σ is the population standard deviation, and n is the sample size.
4. Rearrange the formula to solve for the sample size n: \( n = \left( \frac{z \cdot \sigma}{E} \right)^2 \). Substitute the values for z, σ, and E (2 miles) into the formula.
5. Perform the calculations to determine the minimum sample size n. Round up to the nearest whole number, as sample size must be an integer.

Key Concepts

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed in a statistical study to ensure that the results are reliable and valid. It is influenced by the desired confidence level, the margin of error, and the population standard deviation. A larger sample size generally leads to more accurate estimates of the population parameters.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within a specified range of the sample statistic. A 99% confidence level indicates that if the same sampling procedure were repeated multiple times, approximately 99% of the calculated confidence intervals would contain the true population mean. This high level of confidence typically requires a larger sample size.

Margin of Error

The margin of error is the range within which the true population parameter is expected to lie, based on the sample statistic. In this context, a margin of error of 2 miles means that the sample mean driving distance should be within 2 miles of the actual population mean. The margin of error is influenced by the sample size and the variability of the data, with smaller margins requiring larger samples.