In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.80, σ = 4.1, E = 2.

Verified step by step guidance

1. Identify the formula for determining the minimum sample size n for estimating a population mean: $n=\frac{{\left(z\right)}^{}}{}$_c2σ2E2, where z_c is the critical z-value, σ is the population standard deviation, and E is the margin of error.
2. Determine the critical z-value (z_c) for the given confidence level c = 0.80. Use a z-table or statistical software to find the z-value corresponding to the middle 80% of the standard normal distribution.
3. Substitute the given values into the formula: c = 0.80, σ = 4.1, and E = 2. Replace z_c with the critical z-value obtained in the previous step.
4. Simplify the numerator by squaring the critical z-value (z_c) and multiplying it by the square of the population standard deviation (σ²).
5. Divide the result from the numerator by the square of the margin of error (E²) to calculate 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 crucial for achieving a desired level of confidence and margin of error in estimates. The formula often used involves the population standard deviation, the desired confidence level, and the margin of error.

Confidence Level (c)

The confidence level represents the probability that the sample accurately reflects the population parameter within a specified margin of error. A confidence level of 0.80, for instance, indicates that if the same sampling procedure were repeated multiple times, approximately 80% of the intervals would contain the true population parameter. This concept is essential for understanding the reliability of the estimate.

Margin of Error (E)

The margin of error is the range within which the true population parameter is expected to fall, given a certain confidence level. It quantifies the uncertainty associated with the sample estimate. A smaller margin of error requires a larger sample size, as it indicates a more precise estimate of the population parameter, which is critical for making informed decisions based on the data.