91Ó°ÊÓ

Linear regression is one of the simplest forms of mathematical modeling used to explore the relationship between two or more variables. In multiple regression, multiple predictors are used to estimate the value of a dependent variable. The goal is to determine how well these predictor variables can explain variations in the dependent variable, which, in this case, is the total daily travel time for a truck driver.
The linear equation provided to us is: \( Y = -0.800 + 0.060 x_{1} + 0.900 x_{2} + \epsilon \). Here, \( Y \) is the daily travel time in hours, while \( x_{1} \) and \( x_{2} \) are the predictors - distance traveled and the number of deliveries made, respectively.
In this context, the regression equation allows us to estimate travel time based on specific values of distance and deliveries. By substituting known values into this equation, we can calculate predicted outcomes for various scenarios.

Predictors are independent variables used in a regression model that help explain changes in the dependent variable. In our model, the predictors are:\( x_{1} = \) distance traveled and \( x_{2} = \) number of deliveries. Each has an associated coefficient - a numerical value showing the impact on the dependent variable.
The coefficients \( \beta_{1} = 0.060 \) and \( \beta_{2} = 0.900 \) represent the rate of change in travel time with each unit increase in the predictors. The intercept term \( \beta_{0} = -0.800 \) is the baseline value of \( Y \) when both predictor values are zero. Understanding these coefficients helps in predicting how manipulation of predictors affects the outcome variable.

A z-score, or standard score, measures how a particular value compares to the mean of a set of values, adjusted for standard deviation. Calculating a z-score can tell us how far, and in what direction, a particular value deviates from the mean in terms of standard deviations.
In this exercise, to find the probability of travel time being at most 6 hours, we calculated the z-score using the formula:
\[ z = \frac{(X - \bar{Y})}{\sigma} = \frac{(6 - 4.9)}{0.5} = 2.2 \]
The value 6 is the travel time we are interested in, \( \bar{Y} = 4.9 \) is the mean travel time, and \( \sigma = 0.5 \) is the standard deviation. The z-score tells us the standard distance from the mean.

Once we have a z-score, we can estimate the probability of observing a specific outcome within our data distribution. This involves checking standard normal distribution tables or using statistical software to find the probability corresponding to the z-score.
In this case, a z-score of 2.2 was calculated. Consulting a z-score table, we find that \( P(Z \leq 2.2) = 0.9861 \), indicating there is roughly a 98.61% probability that the driver's travel time will be at most 6 hours.
This high probability suggests that, given the model and provided conditions (50 miles traveled and 3 deliveries), a travel time of 6 hours or less is highly likely.

Understanding what the coefficients indicate is crucial in regression analysis. Coefficients inform us about the direction and magnitude of the relationship between predictors and the dependent variable.
The coefficient \( \beta_{1} = 0.060 \) signifies that for each additional mile traveled, the daily travel time increases by 0.060 hours, assuming the number of deliveries is constant. Meanwhile, \( \beta_{2} = 0.900 \) indicates that each additional delivery adds 0.900 hours to the travel time, assuming the distance traveled remains constant.
These coefficients are pivotal for interpreting how each predictor independently affects the outcome.)