Interval estimation and confidence intervals are crucial tools in statistics for making informed decisions based on sample data. They provide a range of likely values for population parameters, helping engineers assess the reliability of their estimates and make sound judgments.

This section explores how to construct and interpret confidence intervals, determine appropriate sample sizes, and balance confidence levels with interval widths. It also delves into practical applications in engineering, emphasizing the importance of these techniques in real-world problem-solving and decision-making processes.

Confidence Intervals for Parameters

Constructing Confidence Intervals

• A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence
• The level of confidence, typically expressed as a percentage (95%), represents the probability that the interval will contain the true population parameter if the sampling process is repeated many times
• The general formula for a confidence interval is: point estimate ± margin of error, where the margin of error is calculated using the standard error and a critical value from the appropriate distribution (t-distribution or z-distribution)
• To construct a confidence interval for a population mean with a known population standard deviation, use the z-distribution and the formula: $x̄ ± z(α/2) (σ / √n)$, where $x̄$ is the sample mean, $z(α/2)$ is the critical value from the standard normal distribution, $σ$ is the population standard deviation, and $n$ is the sample size

Interpreting Confidence Intervals

• To construct a confidence interval for a population mean with an unknown population standard deviation, use the t-distribution and the formula: $x̄ ± t(α/2, n-1) (s / √n)$, where $s$ is the sample standard deviation and $t(α/2, n-1)$ is the critical value from the t-distribution with $n-1$ degrees of freedom
• Confidence intervals can also be constructed for other population parameters, such as proportions, variances, and differences between means or proportions, using appropriate formulas and distributions
• Interpreting a confidence interval involves understanding that the interval provides a range of plausible values for the population parameter, and that the level of confidence represents the long-run probability of the interval containing the true parameter value
• For example, a 95% confidence interval for a population mean of 50 ± 5 indicates that if the sampling process were repeated many times, approximately 95% of the resulting intervals would contain the true population mean

Sample Size for Confidence Intervals

Factors Influencing Sample Size

• The width of a confidence interval is influenced by the sample size, with larger sample sizes generally resulting in narrower intervals and more precise estimates of the population parameter
• To determine the required sample size for a desired confidence interval width, one needs to specify the level of confidence, the desired margin of error (half the width of the interval), and an estimate of the population standard deviation (if unknown)
• For example, to estimate a population mean with a margin of error of ±3 units and a 95% confidence level, a larger sample size would be required compared to a margin of error of ±5 units at the same confidence level

Calculating Required Sample Size

• The formula for calculating the required sample size for a confidence interval for a population mean with a known population standard deviation is: $n = (z(α/2) σ / E)^2$, where $E$ is the desired margin of error
• When the population standard deviation is unknown, use an estimate from a pilot study or a similar study, and replace $z(α/2)$ with $t(α/2, n-1)$ in the sample size formula. An iterative approach may be necessary to determine the final sample size
• For confidence intervals involving proportions, the required sample size formula is: $n = (z(α/2)^2 * p * (1-p)) / E^2$, where $p$ is an estimate of the population proportion
• For instance, to estimate the proportion of defective items in a production process with a margin of error of ±0.05 and a 95% confidence level, assuming an estimated proportion of 0.1, the required sample size would be: $n = (1.96^2 * 0.1 * 0.9) / 0.05^2 ≈ 139$

Confidence Level, Sample Size, and Interval Width

Relationship between Confidence Level and Interval Width

• Increasing the confidence level (from 90% to 95%) while keeping the sample size constant will result in a wider confidence interval, as a higher level of confidence requires a larger margin of error to account for the increased probability of capturing the true population parameter
• For example, a 90% confidence interval for a population mean may be 50 ± 4, while a 95% confidence interval with the same sample size may be 50 ± 5

Relationship between Sample Size and Interval Width

• Increasing the sample size while keeping the confidence level constant will result in a narrower confidence interval, as a larger sample size reduces the standard error and, consequently, the margin of error
• To maintain a desired confidence interval width while increasing the confidence level, one must increase the sample size to compensate for the larger margin of error required by the higher confidence level
• For instance, to maintain a margin of error of ±3 units when increasing the confidence level from 90% to 95%, the sample size would need to be increased to counteract the wider interval resulting from the higher confidence level

Balancing Confidence Level, Sample Size, and Interval Width

• The relationship between confidence level, sample size, and interval width is crucial for designing studies and determining the resources needed to obtain estimates with the desired level of precision
• Researchers must strike a balance between the desired confidence level, the available resources for sampling, and the acceptable width of the confidence interval
• In practice, this often involves iteratively adjusting the sample size and confidence level until a satisfactory combination is found that meets the study's objectives and constraints

Interval Estimation in Engineering

Identifying Parameters and Collecting Data

• Identify the population parameter of interest (mean, proportion, variance) and determine whether the population standard deviation is known or unknown
• Collect a representative sample from the population, ensuring that the sample size is sufficient for the desired level of confidence and precision
• For example, when estimating the average tensile strength of a material, determine whether the population standard deviation is known from previous studies or if it needs to be estimated from the sample data

Constructing and Interpreting Confidence Intervals

• Calculate the appropriate sample statistics (sample mean, sample proportion, sample variance) and use them to construct the confidence interval using the appropriate formula and distribution
• Interpret the confidence interval in the context of the engineering problem, considering the practical implications of the range of plausible values for the population parameter
• For instance, a 95% confidence interval for the mean tensile strength of a material of 200 MPa ± 10 MPa suggests that the true population mean is likely to fall within this range, which may help determine if the material meets the required specifications

Decision Making and Further Analysis

• Use the confidence interval to make decisions or draw conclusions about the population, such as determining whether a process meets specifications, comparing different designs, or assessing the reliability of a system
• If necessary, use the confidence interval to guide further data collection or experimentation, such as increasing the sample size to achieve a narrower interval or investigating factors that may influence the variability of the parameter
• For example, if the confidence interval for the mean tensile strength of a material is too wide to make a definitive conclusion about its suitability, additional samples may be collected to reduce the margin of error and improve the precision of the estimate

Communicating Results

• Communicate the results of the interval estimation analysis to stakeholders, explaining the limitations and assumptions of the approach and providing recommendations based on the findings
• Present the confidence interval along with its interpretation, the level of confidence, and any relevant context or caveats
• For instance, when reporting the results of a study on the average efficiency of a new engine design, present the confidence interval, explain the practical implications of the range of plausible values, and discuss any limitations or assumptions of the analysis, such as the representativeness of the sample or the potential for measurement error