Sample Size for Mean Find the sample size required to estimate the mean IQ of airline pilots. Assume that we want 99% confidence that the mean from the sample is within two IQ points of the true population mean. Also assume that sigma=15

Verified step by step guidance

1. Step 1: Identify the formula for sample size estimation when estimating a population mean. The formula is: $n\; =\; (z\; *\; \sigma \; /\; E)2$, where n is the sample size, z is the z-score corresponding to the confidence level, σ is the population standard deviation, and E is the margin of error.
2. Step 2: Determine the z-score for a 99% confidence level. For a 99% confidence level, the z-score corresponds to the critical value from the standard normal distribution. This value is approximately 2.576.
3. Step 3: Substitute the given values into the formula. Here, σ = 15 (population standard deviation) and E = 2 (margin of error). The formula becomes: $n\; =\; (2.576\; *\; 15\; /\; 2)2$.
4. Step 4: Simplify the expression inside the parentheses. First, calculate the product of the z-score and the standard deviation, then divide by the margin of error.
5. Step 5: Square the result obtained in Step 4 to find the sample size. Round up to the nearest whole number if necessary, as sample size must be an integer.

Key Concepts

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed in a study to achieve a desired level of precision. It involves considering factors such as the population standard deviation, the desired confidence level, and the margin of error. In this case, we need to calculate how many airline pilots' IQs should be sampled to ensure that our estimate is accurate within two points of the true mean.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within a specified range of the sample estimate. A 99% confidence level indicates that if we were to take many samples, approximately 99% of the calculated confidence intervals would contain the true population mean. This high level of confidence requires a larger sample size to ensure that the estimate is reliable.

Margin of Error

The margin of error is the range within which the true population parameter is expected to fall, given a certain level of confidence. In this scenario, a margin of error of two IQ points means that we want our sample mean to be within two points of the actual mean IQ of airline pilots. This concept is crucial for determining how precise our estimate needs to be and directly influences the required sample size.