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When conducting a survey or experiment, the confidence level tells us how certain we are that our sample mean (or other estimate) falls within a specified range of the true population parameter. For instance, a 99% confidence level implies that if we were to take 100 different samples, approximately 99 of those samples' confidence intervals would contain the true population mean. This gives a high degree of certainty about the estimate.
Confidence level is connected to the critical value and the margin of error. Higher confidence levels require larger sample sizes to ensure the same margin of error, since we need to be more certain our estimate is close to the true value.

The margin of error (E) in a statistical study measures the maximum expected difference between the true population parameter and a sample estimate. It defines the range within which we expect the true population parameter to lie.
In our example, we want the mean IQ score of college professors to be within 4 IQ points of the true mean with 99% confidence. This means our sample mean's range will be up to 4 points away from the true mean in either direction. Having a smaller margin of error requires a larger sample size because it reduces the range in which the true population parameter is expected to fall.
The formula for margin of error, when using the standard normal distribution, often involves the product of the critical value and the standard deviation, divided by the square root of the sample size.

Standard deviation (σ) is a measure of the spread of a set of values. It tells us how much the values deviate on average from the mean. A larger standard deviation indicates greater variability, while a smaller one indicates that values are more closely clustered around the mean.
In the context of the Wechsler IQ test, the standard deviation is given as 15 for the entire population. For a group like college professors, the standard deviation is expected to be smaller due to lesser variation. However, using a higher standard deviation like 15 can ensure our sample size calculation remains conservative, potentially overestimating rather than underestimating the necessary sample size. Using \(\sigma=15\, \) the formula \(\sqrt{(z \times \sigma)\) divides the total variability by the square of the desired precision.

The Wechsler IQ test is a widely used intelligence test designed to measure cognitive ability in adults and older adolescents. It has a mean score set to 100 and a standard deviation of 15. These parameters make it a benchmark for other measurements and studies related to IQ among different populations.
The test assesses various aspects of intelligence through different subtests, including arithmetic, vocabulary, and pattern recognition. Due to these standards, the Wechsler IQ test is often used in academic and psychological research.
In our problem, the specific population of college professors is expected to have a mean IQ above 100, and the standard deviation is assumed to be less than 15, ensuring a more homogenous group with higher cognitive abilities.