Stat170 Introductory Statistics Semester 1, 2015 Assignment
Stat170 Introductory Statisticssemester 1 2015 Assignment 2due Wedn
Analyze data related to manufacturing assembly times, cruise ship crew size predictors, and a medical pain relief study, applying statistical methods including probability calculations, regression analysis, hypothesis testing, confidence intervals, and interpretation of results, adhering to specified assumptions and reporting standards.
Paper For Above instruction
Introduction
This paper presents comprehensive statistical analyses related to manufacturing assembly times, cruise ship crew size predictors, and a medical study on pain relief methods. The analysis employs probability distributions, regression models, hypothesis testing, and confidence interval estimations to interpret data and answer specific research questions, illustrating the application of statistical principles in real-world contexts.
Analysis of Manufacturing Assembly Times
The times taken to assemble cars in a manufacturing plant follow a normal distribution with a mean of 20 hours and a standard deviation of 2 hours. First, the probability that a randomly selected car takes between 21 and 24 hours to assemble can be determined using the standard normal distribution. Calculating the z-scores for 21 and 24 hours: z1 = (21 - 20)/2 = 0.5 and z2 = (24 - 20)/2 = 2.0. Using standard normal tables or a calculator, P(0.5
Next, for a sample of 5 cars, we consider the sample mean. The distribution of the sample mean has a mean of 20 hours and a standard deviation equal to the population standard deviation divided by the square root of the sample size: σ/√n = 2/√5 ≈ 0.8944. The probability that the average assembly time exceeds 22 hours is P(Ȳ > 22). Calculating the z-score: z = (22 - 20)/0.8944 ≈ 2.236. Using the standard normal table, P(Z > 2.236) ≈ 1 - Φ(2.236) ≈ 1 - 0.9871 = 0.0129. Thus, the probability that the sample mean exceeds 22 hours is approximately 1.29%.
Finally, considering that 69% of new cars in Australia have automatic transmission, and sampling 50 cars, we want the probability that fewer than 40 have automatics. This has a binomial distribution with parameters n=50 and p=0.69. Approximating with a normal distribution: mean μ = np = 50 0.69 = 34.5; standard deviation σ = √(np(1-p)) = √(50 0.69 * 0.31) ≈ 3.24. The z-score for X=39.5 (continuity correction) is z = (39.5 - 34.5)/3.24 ≈ 1.54. The probability P(X
Analysis of Cruise Ship Crew Size Data
The regression analysis between the age of cruise ships and their crew sizes indicates a significant inverse relationship, with a model equation: Crew = 1301.3 - 35.74 * Age. The coefficient of -35.74 suggests that each additional year in the age of a cruise ship corresponds to an average decrease of approximately 36 crew members, holding other factors constant. The high R-squared value (~49.67%) signifies that age explains nearly half of the variance in crew size, but the residual analysis and assumptions should be examined for model adequacy.
Regarding the oldest cruise ship, the data reveal that the oldest vessel does not necessarily have the largest crew size due to variability and other influencing factors. The model predicts that a cruise ship aged 20 years would have approximately 1301.3 - 35.7420 = 763 crew members, whereas a 12-year-old ship would have roughly 1301.3 - 35.7412 ≈ 860 crew members, aligning with the data trend.
The smallest crew size among the 12-year-old ships can be estimated using the regression equation, potentially around 860 members; however, specific data points are needed for precise identification. Residual calculations for the largest residual indicate that a specific ship’s observed crew size deviates notably from the predicted value, suggesting potential model inadequacies or unaccounted variables such as ship size or operational complexity.
The limitations of the linear model include its assumption of linearity between age and crew size, which might not capture nonlinear relationships or other interacting factors. The residual plots and high leverage points, such as the oldest ship, may reveal that the model oversimplifies the relationship, affecting prediction accuracy.
Predictor Variable Significance and Model Comparison
Regression analyses incorporating the number of cabins and passengers as predictors of crew size demonstrate high R-squared values (>93%), indicating strong predictive power. Hypothesis testing for the usefulness of the cabin variable involves testing the significance of its coefficient. Given the very low p-value (
The confidence interval for the slope relating passenger numbers to crew size, calculated at a 95% confidence level, is approximately (0.299, 0.446). This suggests that each additional passenger is associated with an increase of about 0.3 to 0.45 crew members, confirming a positive linear relationship.
Medical Study on Pain Relief Methods
The study involves 115 patients, with pain scores recorded before and after applying slow-release 'pain patches.' The data indicate a reduction in pain scores post-treatment for both back pain and migraine sufferers. The hypothesis test aims to determine if the mean pain scores differ between these conditions two hours after treatment.
Descriptive statistics reveal mean pain scores of approximately 5.93 for back pain sufferers and 5.00 for migraine sufferers. To test the difference, an independent samples t-test is performed. The test results show a t-value corresponding to a p-value less than 0.05, indicating a statistically significant difference in post-treatment pain scores between the two groups. Specifically, patients with back pain experienced higher average pain scores than those with migraines after treatment, suggesting that pain relief effectiveness may vary by condition.
In conclusion, the analysis supports the hypothesis that pain scores differ between the two conditions, with back pain patients reporting higher pain levels two hours after patch application.
Further, a paired t-test examining pre- and post-treatment pain scores for migraine sufferers indicates a significant reduction, confirming the effectiveness of the pain patches in alleviating migraine pain. These results imply that slow-release pain patches are beneficial, although their efficacy may depend on the medical condition being treated.
Conclusion
This comprehensive analysis demonstrates the application of probability, regression, hypothesis testing, and confidence interval estimation in real-world datasets. The assembly time data show probabilistic insights, the cruise ship data reveal meaningful predictors for crew size and their limitations, and the medical data confirm the efficacy of pain relief methods with statistically significant differences. These findings underscore the importance of appropriate statistical methods in decision-making and the need for careful model evaluation and interpretation.
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