Mean IQ of Data Scientists See the preceding exercise, in which we can assume that sigma=15 for the IQ scores. Data scientists are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of data scientists, given that we want 98% confidence that the sample mean is within 2 IQ points of the population mean. Does the sample size appear to be practical?

Verified step by step guidance

1. Identify the formula for determining the required sample size for estimating a population mean: \( n = \left( \frac{z \cdot \sigma}{E} \right)^2 \), where \( n \) is the sample size, \( z \) is the z-score corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( E \) is the margin of error.
2. Determine the values for the variables in the formula: \( \sigma = 15 \) (given), \( E = 2 \) (the desired margin of error), and the confidence level is 98%, which corresponds to a z-score of approximately 2.33 (from a z-table or standard normal distribution).
3. Substitute the known values into the formula: \( n = \left( \frac{2.33 \cdot 15}{2} \right)^2 \).
4. Simplify the expression inside the parentheses: \( \frac{2.33 \cdot 15}{2} \), then square the result to calculate \( n \).
5. Evaluate whether the resulting sample size is practical by considering factors such as the feasibility of collecting data from the calculated number of participants and the resources available for the study.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In this context, a 98% confidence level indicates that if we were to take many samples, approximately 98% of the calculated intervals would contain the true mean IQ of data scientists. This concept is crucial for understanding how sample size affects the precision of our estimates.

Sample Size Calculation

Sample size calculation involves determining the number of observations needed to achieve a desired level of precision in estimating a population parameter. The formula typically incorporates the desired confidence level, the population standard deviation (sigma), and the margin of error. In this case, we need to calculate how many data scientists' IQ scores are required to ensure that our estimate is within 2 IQ points of the true mean with 98% confidence.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, the population standard deviation (sigma = 15) indicates how much individual IQ scores of data scientists deviate from the mean. Understanding standard deviation is essential for calculating the sample size, as it influences the width of the confidence interval and the required sample size for a given margin of error.