In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.90, σ = 6.8, E = 1.

Verified step by step guidance

1. Identify the formula for determining the minimum sample size n for estimating a population mean: $n=(\frac{z}{}$_c⋅σE)2, 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 = 0.90. Use a z-table or statistical software to find the z-value corresponding to the middle 90% of the standard normal distribution.
3. Substitute the given values into the formula: σ = 6.8 and E = 1. Replace z_c with the critical z-value obtained in the previous step.
4. Simplify the fraction $\frac{z}{}$_c⋅σE by multiplying z_c and σ, then dividing by E.
5. Square the result of the fraction to compute the minimum sample size n. If n is not a whole number, always round up to the nearest integer, 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 confidence interval will contain the true population parameter. A confidence level of 0.90 indicates that there is a 90% chance that the interval estimate will capture the true value. This level influences the width of the confidence interval and, consequently, the required sample size.

Margin of Error (E)

The margin of error is the range within which the true population parameter is expected to lie, given a certain confidence level. It quantifies the uncertainty associated with sample estimates. A smaller margin of error requires a larger sample size, as it indicates a desire for more precise estimates of the population parameter.