Estimation of a Population Mean: Margin of Error and Sample Size Determination

Confidence Intervals for One Population Mean

Interpreting a Confidence Interval

A confidence interval provides a range of plausible values for a population parameter, such as the mean. It is constructed so that, with a specified level of confidence (e.g., 95%), the interval will contain the true population mean in repeated sampling.

• Correct Interpretation: A 95% confidence interval for the mean height of all adult American men, calculated as (68.91, 69.69), means we are 95% confident that the mean height of all adult American men is between 68.91 and 69.69 inches.

• Incorrect Interpretation: It does not mean that 95% of individual men are between these heights; rather, it refers to the population mean.

Key Point: Always state that the confidence interval is about the population mean, not individual values.

Margin of Error and Its Control

The margin of error (E) quantifies the maximum expected difference between the sample mean and the true population mean at a given confidence level. A smaller margin of error means a more precise estimate.

• The formula for the margin of error when the population standard deviation (σ) is known is:

$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

• zα/2: Critical value from the standard normal distribution for the desired confidence level (e.g., 1.96 for 95%).

• σ: Population standard deviation.

• n: Sample size.

If the margin of error is too large for practical purposes (e.g., you want to estimate the mean to within 0.1 inch), you can adjust the sample size to achieve the desired precision.

Determining Required Sample Size for a Desired Margin of Error

To ensure the margin of error does not exceed a specified value (E), solve the margin of error formula for n:

$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2$

• Example: Suppose you want a 95% confidence interval for the mean height of adult American men with a margin of error no greater than 0.1 inch. Given σ = 2.8 and zα/2 = 1.96:

$n = \left( \frac{1.96 \cdot 2.8}{0.1} \right)^2 = (54.88)^2 = 3011.44$

• Since sample size must be a whole number, always round up to the next integer, regardless of the decimal part. Thus, n = 3012.

• Reason: Rounding down would result in a margin of error larger than desired, which is unacceptable. Rounding up ensures the margin of error is within the specified limit.

Effect of Sample Size on Margin of Error

• As the sample size (n) increases, the margin of error (E) decreases, making the estimate more precise.

• Conversely, a small sample size leads to a larger margin of error and less precise estimates.

• This relationship is intuitive: larger samples better represent the population, reducing uncertainty.

Summary Table: Sample Size Determination

Parameter	Symbol	Description
Confidence Level	1 - \alpha	Probability the interval contains the true mean (e.g., 0.95)
Critical Value	z_{\alpha/2}	Value from standard normal table (e.g., 1.96 for 95%)
Population Standard Deviation	\sigma	Known or estimated variability in the population
Margin of Error	E	Maximum acceptable difference between sample and population mean
Sample Size	n	Number of observations required (always round up)

Key Takeaways

• Confidence intervals estimate population parameters with a specified level of confidence.

• The margin of error depends on the confidence level, population standard deviation, and sample size.

• To achieve a smaller margin of error, increase the sample size.

• When calculating required sample size, always round up to ensure the margin of error does not exceed the desired value.