FINDING SAMPLE SIZE Instead of using Table 7-2 for determining the sample size required to estimate a population standard deviation σ, the following formula can also be used


$n=\frac{1}{2}{\left(\frac{z_{\alpha /2}}{d}\right)}^2$


where $z_{{}_{}\alpha /2}$ corresponds to the confidence level and d is the decimal form of the percentage error. For example, to be 95% confident that s is within 15% of the value of σ, use zα/2=1.96 and d=0.15 to get a sample size of n=86. Find the sample size required to estimate the standard deviation of IQ scores of data scientists, assuming that we want 98% confidence that s is within 5% of σ.

Verified step by step guidance

1. Step 1: Understand the formula for determining the sample size: n = (1/2) * (zα/2 / d)^2. Here, zα/2 corresponds to the z-score for the desired confidence level, and d is the decimal form of the percentage error.
2. Step 2: Identify the given values from the problem. For a 98% confidence level, zα/2 = 2.33 (this is the z-score corresponding to 98% confidence). The percentage error is 5%, so d = 0.05.
3. Step 3: Substitute the values of zα/2 and d into the formula. The formula becomes n = (1/2) * (2.33 / 0.05)^2.
4. Step 4: Simplify the expression inside the parentheses first. Calculate (2.33 / 0.05), then square the result.
5. Step 5: Multiply the squared result by (1/2) to find the sample size n. This will give you the required sample size to estimate the standard deviation with the specified confidence level and error margin.

Key Concepts

Sample Size Determination

Sample size determination is a statistical process used to calculate the number of observations or replicates needed in a study to ensure that the results are reliable and valid. It is crucial for achieving a desired level of precision in estimating population parameters, such as the standard deviation. The formula provided in the question helps to find the minimum sample size required to estimate the population standard deviation with a specified confidence level and margin of error.

Confidence Level

The confidence level represents the degree of certainty that a population parameter lies within a specified interval. It is typically expressed as a percentage, such as 95% or 98%, indicating the proportion of times that the confidence interval would contain the true parameter if the study were repeated multiple times. In the context of the question, a 98% confidence level means that there is a 2% chance that the true standard deviation falls outside the calculated interval.

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. It is often expressed as a percentage of the estimated value, indicating how much the estimate could vary. In the question, a margin of error of 5% means that the sample standard deviation should be within 5% of the actual population standard deviation, which directly influences the required sample size for accurate estimation.