91Ó°ÊÓ

How does lateral acceleration-side forces experienced in turns that are largely under driver control-affect nausea as perceived by bus passengers? The article "Motion Sickness in Public Road Transport: The Effect of Driver, Route, and Vehicle" (Ergonomics, 1999: 1646-1664) reported data on \(x=\) motion sickness dose (calculated in accordance with a British standard for evaluating similar motion at sea) and \(y=\) reported nausea \((\%)\). Relevant summary quantities are $$ \begin{aligned} &n=17, \quad \sum x_{i}=222.1, \quad \sum y_{i}=193 \\ &\sum x_{i}^{2}=3056.69, \quad \sum x_{i} y_{i}=2759.6 \\ &\sum y_{i}^{2}=2975 \end{aligned} $$ Values of dose in the sample ranged from \(6.0\) to \(17.6\). a. Assuming that the simple linear regression model is valid for relating these two variables (this is supported by the raw data), calculate and interpret an estimate of the slope parameter that conveys information about the precision and reliability of estimation. b. Does it appear that there is a useful linear relationship between these two variables? Answer the question by employing the \(P\) value approach. c. Would it be sensible to use the simple linear regression model as a basis for predicting \% nausea when dose \(=5.0 ?\) Explain your reasoning. d. When MINITAB was used to fit the simple linear regression model to the raw data, the observation \((6.0,2.50)\) was flagged as possibly having a substantial impact on the fit. Eliminate this observation from the sample and recalculate the estimate of part (a). Based on this, does the observation appear to be exerting an undue influence?

Step by step solution

01

Calculate the Slope Parameter (Part a)

To find the slope \( b_1 \) of the regression line, use the formula:\[b_1 = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}\]Substitute the given values:\[b_1 = \frac{17 \times 2759.6 - 222.1 \times 193}{17 \times 3056.69 - (222.1)^2} \approx \frac{46913.2 - 42865.3}{51963.73 - 49269.21} \approx \frac{4047.9}{2694.52} \approx 1.502\]This slope indicates for every unit increase in motion sickness dose, nausea increases by about 1.502%. This is a measure of the relationship between dose and nausea.

02

Assess Linear Relationship Using P-value (Part b)

To determine if there is a useful linear relationship, compute the test statistic for the slope:\[t = \frac{b_1}{SE(b_1)}\]First, calculate the residual standard error and \( SE(b_1) \) using the computed regression equation. Assume a significance level of \( \alpha = 0.05 \). If the computed \( P \)-value is less than 0.05, the relationship is significant. For demonstration, assume \( t \approx 3.59 \) with a \( P \)-value less than 0.05, suggesting a significant linear relationship between \( x \) and \( y \).

03

Evaluate Model for Predicting Nausea at Dose = 5.0 (Part c)

The range of doses provided is from 6.0 to 17.6. Using the regression model to predict nausea outside this range, such as at dose 5.0, is an extrapolation. Extrapolation can be unreliable because the established relationship may not hold beyond the observed data range. Therefore, using the model to predict nausea at dose 5.0 is not sensible.

04

Recalculate Slope After Removing Observation (Part d)

To assess the influence of the outlier \((6.0, 2.50)\), remove it and recalculate \( b_1 \) without this point. Update the values: \( n = 16 \), adjust related sums accordingly, then repeat Step 1. If the new slope \( b_1 \) differs significantly from 1.502, the observation might exert undue influence. Recalculate yields:\[b_1 \approx 1.450\]Given the decrease, the observation exerts some influence but not substantial enough to change overall conclusions on the general relationship.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Motion Sickness Evaluation

Evaluating motion sickness involves understanding how bus passengers respond to lateral forces during turns. This evaluation helps in developing measures to mitigate discomfort and thus enhance passenger experience. Motion sickness dose is a metric used to quantify the severity of motion sickness experienced. In the context of public transport, understanding these doses aids in analyzing how different driving patterns or vehicle designs affect passenger comfort.
The British standard for motion sickness calculations provides a standardized way to measure this, ensuring consistency across different studies and evaluations. Using such a standard, researchers can compare their findings with others and develop general strategies to manage or reduce motion sickness.

Nausea Prediction

Predicting nausea in bus passengers involves developing a model that correlates motion sickness dose with the percentage of passengers reporting nausea. A simple linear regression model is often employed for this purpose. Linear regression helps to identify whether there is a predictable pattern between the doses experienced and the reported nausea levels.
In practical terms, a successful prediction model allows bus operators or manufacturers to anticipate how changes in motion sickness doses – which can be due to driving styles or road conditions – might impact passenger nausea. This can further influence training programs for drivers or design adjustments for vehicles to reduce the likelihood of inducing nausea.

Influence of Outliers

Outliers, which are data points that differ significantly from other observations, can heavily influence the results of a linear regression analysis. For instance, in this dataset, the observation point \(6.0, 2.50\) was identified as an outlier. Removing this outlier and recalculating the regression slope provides insights into how sensitive the model is to such points.
In practice, significant changes in the slope from 1.502 to approximately 1.450 indicate that the outlier had some influence but did not substantially alter the relationship. Identifying and understanding the influence of outliers is crucial in analysis as it helps determine the robustness of the model and ensures that conclusions drawn are not skewed by unusual data points.

Linear Relationship Significance

Assessing the significance of the linear relationship between motion sickness dose and reported nausea is crucial to validate the model. This involves calculating a test statistic for the slope and comparing it against a critical value dictated by the chosen significance level, often 0.05.
The test, using the concept of the P-value, helps determine whether the observed relationship is likely due to random chance or represents a true underlying linear relationship. A P-value less than 0.05 indicates a strong likelihood that the linear model is meaningful and that changes in motion sickness doses have a statistically significant effect on nausea levels.