Book On Statistics And Probability For Engineering ✓ Solved

Book Statistics And Probability With Applications For Engineers And S

Analyze and interpret various statistical techniques and their applications in engineering context, including trend forecasting with exponential smoothing, trend analysis, model comparison, nonparametric testing assumptions, Kruskal-Wallis test conditions, hypothesis testing for treatment effects, restaurant rating comparisons, nonparametric paired difference tests, and correlation analysis between exam scores. Address each question by applying appropriate statistical formulas, methods, and interpretation of results, emphasizing the underlying assumptions, model effectiveness (e.g., MAD criterion), and relevant statistical theory.

Sample Paper For Above instruction

In the realm of engineering and scientific research, statistical analysis plays a vital role in understanding data patterns, making forecasts, comparing groups, and testing hypotheses under conditions where classical parametric assumptions may not hold. This paper meticulously examines various statistical methods as rooted in the provided context, offering a comprehensive interpretation aligned with real-world applications.

Forecasting Tourist Stay Using Exponential Smoothing

The initial task involves applying exponential smoothing with a smoothing constant (w) of 0.6 to forecast the average nights foreign tourists spend in Washington DC for four upcoming quarters in 2017. Exponential smoothing is a versatile forecasting technique that assigns exponentially decreasing weights to past observations, effectively capturing recent trends and patterns (Holt, 1957). The formula for simple exponential smoothing is:

\( S_t = \alpha \times X_t + (1 - \alpha) \times S_{t-1} \)

where \( S_t \) indicates the forecasted value at time t, \( X_t \) the actual observed value, and \( \alpha \) the smoothing constant (in this case, 0.6).

Using historical quarterly data, the forecast for the next period can be computed iteratively. Starting with the initial value \( S_1 \), typically the first observed value or an average of initial data points, subsequent forecasts are generated. This method effectively accounts for recent fluctuations, especially when the smoothing factor is high (0.6), which emphasizes recent data.

To incorporate potential trend components, an advanced form such as Holt’s linear trend method can be employed, which includes a trend component \( T_t \). The equations are:

\( S_t = \alpha X_t + (1 - \alpha)(S_{t-1} + T_{t-1}) \)

\( T_t = \beta (S_t - S_{t-1}) + (1 - \beta) T_{t-1} \)

where \( \beta \) is the trend smoothing coefficient, set here as 0.2. Using this method, the forecasted stay combines level and trend extrapolation, allowing for more nuanced projections that can reflect underlying data trends (Chatfield, 2000).

Forecasts are then generated for each quarter of 2017, providing valuable insights into future tourism patterns.

Model Comparison Using MAD Criterion

Two forecasting models are considered: one based solely on exponential smoothing and another incorporating a trend component with smoothing parameters \( \alpha = 0.6 \) and \( \nu = 0.2 \). Evaluating model performance involves calculating the Mean Absolute Deviation (MAD), which measures the average absolute error between forecasted and actual values. A lower MAD indicates superior predictive accuracy and model reliability (Hyndman & Athanasopoulos, 2018).

Suppose the MADs for both models have been computed from historical data; the model with the smaller MAD is preferred. The trend-adjusted model generally performs better if the data exhibit a notable trend component, as it captures the directional movement more effectively than the simple exponential smoothing model.

In this context, assuming the trend model exhibits a lower MAD, it would be considered the more appropriate choice, confirming the presence of an underlying trend in the tourist stay data.

Assumptions Underlying Kruskal-Wallis Test

The nonparametric Kruskal-Wallis test assesses whether multiple samples originate from the same distribution without assuming normality. Its key assumptions include:

• Sampled data are independent within and across groups.
• The measurement scale is at least ordinal.
• Samples are randomly drawn from the population.

Option C correctly identifies these assumptions: I. Random sampling, II. Independence of cases, and III. Ordinal measurement scale (Siegel & Castellan, 1988).

Testing for Differences in Plant Weights Across Treatments

In the case of plant weights under different treatments (no treatment, fertilizer, irrigation, fertilizer and irrigation), the samples are independent, and the data are not normally distributed. Since the normality assumption is violated, a nonparametric test like Kruskal-Wallis is appropriate. The test compares the distributions across groups to determine whether they differ significantly, using rank-based statistics rather than means (Hollander & Wolfe, 1973).

The null hypothesis states that the distributions are identical across treatments. The alternative suggests at least one treatment yields different plant weights. Conducting the test involves ranking all observations collectively, computing the sum of ranks for each group, and then calculating the test statistic (\( H \)). A significant \( p \)-value (

Based on the data analysis, the conclusion might be that the weights differ significantly among treatments, suggesting the treatments impact plant growth variably. The effectiveness of fertilizers or irrigation can thus be statistically validated using the Kruskal-Wallis test.

Comparing Restaurant Ratings with Friedman Test

The problem involves six critics rating four restaurants, aiming to determine whether there are differences among ratings. Since the same raters evaluate all restaurants, the data are paired, and the nonparametric Friedman test (a variant of repeated measures ANOVA) is suitable (Friedman, 1937). It tests for differences in the rankings among the groups, accounting for the within-subject design.

The test involves ranking the restaurants within each rater, computing the sum of ranks per restaurant, and calculating a chi-square statistic. A significant result indicates differences in restaurant ratings. This objective approach provides evidence to support or refute the hypothesis that all restaurants are rated similarly by critics (Conover, 1999).

Nonparametric Tests for Paired Differences

For analyzing paired data where the differences are not necessarily normally distributed, suitable nonparametric tests include:

• a. Wilcoxon Signed Ranks Test
• b. Sign Test
• c. Kruskal-Wallis test (not for paired data)
• d. Spearman's Rank Correlation (for association, not difference testing)

Hence, the correct options are a. and b., but not c. or d., as they serve different analytical purposes (Hollander & Wolfe, 1973).

Assessing Association Between Math and English Scores

Given the scores of 10 students in Math and English, where the relationship may not be linear, Spearman's rank correlation coefficient (\( \rho \)) is appropriate to measure the strength and direction of the association in a nonparametric context (Spearman, 1904). This statistic assesses how well the relationship between two variables can be described using a monotonic function, making it suitable for non-linear relationships.

Calculating Spearman’s \( \rho \) involves ranking each set of scores, computing the difference in ranks for each student, and then applying:

\( \rho = 1 - \frac{6 \sum d_i^2}{n(n^2 -1)} \)

where \( d_i \) is the difference in ranks, and \( n \) is the number of observations.

A significant correlation coefficient would suggest a monotonic relationship exists, providing insights into how these scores are related across students without assuming linearity.

Conclusion

Statistical methodologies such as exponential smoothing and trend analysis facilitate forecasting in dynamic environments like tourism data. Model comparison criteria like MAD ensure the selection of the most accurate models. Nonparametric tests like Kruskal-Wallis and Friedman are indispensable when classical assumptions are violated, especially with ordinal or skewed data. Tests for associations, exemplified by Spearman’s correlation, enable understanding relationships in non-linear contexts. Overall, careful selection based on underlying data characteristics and test assumptions leads to robust and valid inferences in engineering and scientific analyses.

References

• Chatfield, C. (2000). The Analysis of Time Series: An Introduction, Sixth Edition. Chapman & Hall/CRC.
• Conover, W. J. (1999). Practical Nonparametric Statistics, 3rd Edition. Wiley.
• Friedman, M. (1937). The Use of Ranks of the Data in Analysis of Variance. Journal of the American Statistical Association, 32(200), 675-701.
• Holt, C. C. (1957). Forecasting seasonals and trends by exponentially weighted moving averages. Office of Naval Research.
• Hollander, M., & Wolfe, D. A. (1973). Nonparametric Statistical Methods. Wiley.
• Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and Practice. OTexts.
• Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences, 2nd Edition. McGraw-Hill.
• Spearman, C. (1904). The Proof and Measurement of Association between Two Test Scores. American Journal of Psychology, 15(1), 72-101.