Module 2 Homework Assignment 1: Randomly Selected Light Bulb
Module 2 Homework Assignment1 459 Randomly Selected Light Bulbs Were
Tested in a laboratory, 459 light bulbs were examined, of which 291 lasted more than 500 hours. The assignment involves estimating the true proportion of all light bulbs that last more than 500 hours, finding critical values for confidence intervals, calculating margins of error, constructing confidence intervals, interpreting those intervals, determining when to use Student’s t-distribution, and analyzing sample data related to college students’ earnings.
Paper For Above instruction
This paper aims to address the statistical analysis undertaken in the evaluation of light bulb longevity and related confidence interval estimations, as well as the application of statistical distributions for population means and proportions. The analysis synthesizes concepts from inferential statistics, including point estimation, critical value determination, margin of error calculation, confidence interval construction, and interpretation. This comprehensive review demonstrates how statistical tools assist in making informed conclusions based on sample data, underpinning research and quality assurance processes in practical settings.
Introduction
Statistics serve as a foundational tool in evaluating data reliability and making inferences about larger populations. In practical applications, such as product testing or social science research, statistical techniques enable researchers to estimate parameters, assess the variability of data, and interpret the results within specified confidence levels. This paper examines multiple statistical tasks, beginning with a basic proportion estimate for the duration of light bulbs, progressing through the use of critical values in confidence intervals, and extending to applications involving the t-distribution and analysis of sample means in a different context—college students’ earnings.
Estimating the True Proportion of Long-Lasting Light Bulbs
The primary data point involves 459 light bulbs tested, with 291 lasting more than 500 hours. The point estimate for the proportion of all such bulbs that last beyond this threshold is computed as the ratio of successful outcomes to the total sample size: p̂ = 291/459 ≈ 0.634. This proportion provides a preliminary estimate, suggesting approximately 63.4% of light bulbs can be expected to last more than 500 hours in the population. This estimate sets the stage for further inferential procedures, including confidence interval construction, which quantifies the uncertainty surrounding this proportion estimate.
Critical Value for a 98% Confidence Level
Determining the critical value zα/2 for a confidence level of 98% involves finding the z-score that captures the central 98% of the standard normal distribution. With α = 1 - 0.98 = 0.02, the value of α/2 is 0.01. Consulting standard normal distribution tables or statistical software yields a zα/2 of approximately 2.33. This critical value is essential for constructing confidence intervals, enabling the analyst to specify the range within which the true population parameter is likely to fall with 98% certainty.
Margin of Error for the Proportion with n=163 and x=96
The margin of error (E) quantifies the precision of the estimate and is calculated using the formula: E = zα/2 √[p̂(1 - p̂)/n]. First, the sample proportion is computed as p̂ = 96/163 ≈ 0.589. For a 95% confidence level, zα/2 is 1.96. Substituting values: E = 1.96 √[0.589 (1 - 0.589)/163] ≈ 1.96 √[0.589 0.411/163] ≈ 1.96 √[0.242/163] ≈ 1.96 √0.00149 ≈ 1.96 0.0386 ≈ 0.0756. Therefore, the margin of error is approximately 0.076, indicating the amount the estimate may vary due to sampling variability.
Constructing the Confidence Interval for the Proportion
The confidence interval (CI) is constructed by adding and subtracting the margin of error from the point estimate: CI = p̂ ± E. Using p̂ ≈ 0.589 and E ≈ 0.076, the interval is approximately (0.513, 0.665). This range implies that, with 95% confidence, the true proportion of all light bulbs lasting more than 500 hours is between 51.3% and 66.5%. Such a range provides a quantitative measure of the estimate’s reliability and helps stakeholders understand the variability inherent in sampling processes.
Interpretation of the Confidence Interval
The constructed confidence interval indicates that, based on the sample data, we are 95% confident that the proportion of all light bulbs capable of lasting more than 500 hours lies somewhere between approximately 51% and 67%. This means that if similar sampling methods were repeated multiple times, about 95% of the resulting confidence intervals would contain the true population proportion. It offers a practical understanding of the uncertainty and variability associated with sampling, aiding decision-makers in assessing product quality and reliability.
When to Use Student’s t-Distribution
The Student’s t-distribution is appropriate when estimating a population mean from a small sample size (n
Critical Value for t at n=12 and 95% Confidence
Calculating the critical t-value for a sample of 12 observations at the 95% confidence level involves degrees of freedom df = n - 1 = 11. Using t-distribution tables or software, the tα/2 value for df = 11 at 95% confidence is approximately 2.201. This value is essential for constructing confidence intervals for the mean when the population standard deviation is unknown and the sample size is small.
Analyzing College Students’ Earnings Data
The sample includes 81 students with a mean annual earning of $3,967 and a sample standard deviation of $874. The task involves calculating the margin of error (E) for a 99% confidence interval. The critical t-value for df = 80 (since n = 81) at 99% confidence is approximately 2.634. Applying the formula E = tα/2 (s/√n), we get E ≈ 2.634 (874/√81) ≈ 2.634 (874/9) ≈ 2.634 97.11 ≈ $255.72. This margin of error indicates the degree of uncertainty associated with the estimated mean salary.
Constructing the Confidence Interval for Earnings
The confidence interval is calculated as: CI = x̄ ± E, which yields ($3,967 - $255.72, $3,967 + $255.72), or approximately ($3,711.28, $4,222.72). This interval suggests that, with 99% confidence, the true average annual earnings of college students fall within this range. Such information is valuable for policymakers and educational institutions aiming to evaluate the economic outcomes of students.
Interpretation of the Earnings Confidence Interval
In non-technical terms, we can say that we are 99% confident that the average yearly earnings of college students are between roughly $3,711 and $4,223. This means that if a similar survey were conducted multiple times, approximately 99% of such intervals would contain the true average income. This level of confidence provides a strong indication of the typical earnings expectation for college students, informing decisions made by students, educators, and policy advisors.
Conclusion
The statistical analyses presented demonstrate essential concepts in inferential statistics, including point estimation, critical value determination, margin of error calculation, and confidence interval interpretation. Applying these techniques to real-world data, such as light bulb longevity and student earnings, showcases their practical importance. These methods enable researchers and decision-makers to quantify uncertainty and make evidence-based conclusions, underpinning the reliability and validity of statistical inference in diverse fields.
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