Determining Sample Size The given expression is used to determine the size of the sample necessary to estimate the proportion of college students who have the profound wisdom to take a statistics course. Find the value and round the result up to the next larger whole number.


[(1.95996)^2 - 0.25] / (0.03)^2

Verified step by step guidance

1. Identify the formula for determining the sample size for estimating a proportion, which is given by: \( n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \), where \( Z \) is the Z-score, \( p \) is the estimated proportion, and \( E \) is the margin of error.
2. Recognize that the expression provided is a simplified version of the sample size formula, where \( Z = 1.95996 \), \( p = 0.5 \) (since \( 0.25 = 0.5 \times 0.5 \)), and \( E = 0.03 \).
3. Substitute the given values into the formula: \( n = \frac{(1.95996)^2 \cdot 0.5 \cdot 0.5}{(0.03)^2} \).
4. Calculate the numerator: \( (1.95996)^2 \cdot 0.5 \cdot 0.5 \).
5. Calculate the denominator: \( (0.03)^2 \), then divide the numerator by the denominator and round up to the next whole number to find the required sample size.

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 achieve a desired level of precision. It ensures that the sample accurately reflects the population, minimizing errors and increasing the reliability of the results. In this context, it involves using a formula to estimate the proportion of a population with a specific characteristic.

Confidence Level and Z-Score

The confidence level represents the degree of certainty that the population parameter lies within the confidence interval. A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values, often used in the context of confidence intervals. In the given expression, 1.95996 is the Z-score corresponding to a 95% confidence level, indicating the range within which the true population proportion is expected to fall.

Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It defines the range within which the true population parameter is expected to lie, with a certain level of confidence. In the formula, 0.03 represents the desired margin of error, indicating the acceptable range of deviation from the true proportion of students taking a statistics course.