Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 99% confident that the sample mean is within two units of the population mean and σ = 1.4. Assume the population is normally distributed.

Verified step by step guidance

1. Identify the formula for determining the minimum sample size for estimating a population mean: $n=\left(\frac{z_{\mathrm{\alpha /2}}}{}\right)^2$, where $z_{\mathrm{\alpha /2}}$ is the critical z-value, $\sigma$ is the population standard deviation, and $E$ is the margin of error.
2. Determine the critical z-value for a 99% confidence level. Since the confidence level is 99%, the significance level $\alpha$ is 0.01, and ${\alpha }_2$ is 0.005. Use a z-table or statistical software to find the z-value corresponding to a cumulative probability of 0.995.
3. Substitute the given values into the formula. Here, $\sigma$ = 1.4 and $E$ = 2. The formula becomes: $n=(\frac{z_{\mathrm{\alpha /2}}}{·}/2)^2$.
4. Simplify the expression inside the parentheses by multiplying the critical z-value by the standard deviation and dividing by the margin of error.
5. Square the result from the previous step to calculate the minimum sample size. If the result 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 needed in a sample to achieve a desired level of confidence and precision. It is crucial in statistics to ensure that the sample accurately reflects the population, allowing for valid inferences. The formula often used involves the desired confidence level, the population standard deviation, and the margin of error.

Confidence Level

The confidence level represents the probability that the sample mean will fall within a specified range of the population mean. In this case, a 99% confidence level indicates that if we were to take many samples, 99% of the time, the sample mean would be within two units of the true population mean. This concept is essential for understanding the reliability of the sample estimates.

Margin of Error

The margin of error is the range within which the true population parameter is expected to lie, given a certain level of confidence. In this scenario, a margin of error of two units means that the sample mean should be within two units of the population mean. This concept is vital for determining how precise the sample estimate needs to be in relation to the population parameter.