Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean.
a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams.

Verified step by step guidance

1. Step 1: Recall the formula for determining the minimum sample size required for estimating a population mean: n = (Z * σ / E)^2, where Z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error.
2. Step 2: Identify the given values from the problem: the confidence level is 95%, so the z-score (Z) for a 95% confidence level is approximately 1.96. The population standard deviation (σ) is 3.10 milligrams, and the margin of error (E) is 0.75 milligrams.
3. Step 3: Substitute the given values into the formula: n = (1.96 * 3.10 / 0.75)^2.
4. Step 4: Simplify the expression inside the parentheses first: calculate (1.96 * 3.10) and then divide the result by 0.75.
5. Step 5: Square the result from Step 4 to find the minimum sample size (n). If the result is not a whole number, always round up to the nearest whole number, as sample size must be an integer.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals, approximately 95% of those intervals would contain the true population mean.

Sample Size Determination

Determining the minimum sample size is crucial for ensuring that the estimate of the population mean is accurate within a specified margin of error. The formula for calculating sample size involves the desired confidence level, the population standard deviation, and the margin of error, which in this case is 0.75 milligrams.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this context, the population standard deviation of 3.10 milligrams indicates how much individual cholesterol content values deviate from the mean, which is essential for calculating the sample size needed for the confidence interval.