Mat 117 Spring 2019 Assignment 9 Using MegaState Excel

Mat 117spring 2019assignment 9 Using Megastatexceldr S Sidhom1 A

A random sample of eight auto drivers insured with a company and having similar auto insurance policies was selected. The following table lists their driving experience (in years) and the monthly auto insurance premium (in dollars).

Driving Experience (years): 2, 4, 6, 8, 10, 12, 14, 16

Monthly Auto Insurance Premium ($): 150, 180, 210, 240, 270, 300, 330, 360

This assignment requires performing multiple regression analysis and interpretation based on the data provided. Tasks include calculating the least squares regression model of monthly premiums on driving experience, interpreting the slope, evaluating the significance of the slope, predicting premiums for specific experience levels, calculating and interpreting the correlation coefficient, assessing the significance of the correlation, determining the coefficient of determination, and visually representing the data through a scatter plot with the regression line.

Paper For Above instruction

The relationship between driving experience and auto insurance premiums can be studied effectively through regression analysis. Analyzing this data helps understand how the driving experience influences the monthly insurance premium and whether this relationship is statistically significant. Using Excel or MegaStat, we can derive the regression model, interpret its parameters, evaluate the strength and significance of the relationship, and visualize the findings.

Introduction

Auto insurance premiums vary based on multiple factors, among which driver experience plays a significant role. Typically, more experienced drivers are perceived as less risky, which influences premium costs. To analyze this relationship, regression analysis provides a quantitative way to examine how driving experience affects insurance costs. This paper employs data from a sample of eight drivers, performing various statistical analyses to interpret this association comprehensively.

Regression Analysis and Model Formulation

The first step involves calculating the least squares regression of monthly premiums on driving experience. Given the data points, the regression model follows the general form:

Y = a + bX

where Y represents the monthly premium, X is the driving experience in years, b is the slope, and a is the intercept.

Using Excel or MegaStat, the regression output indicates that the estimated slope (b) is approximately 15, and the intercept (a) is around 120. Therefore, the regression equation is:

Premium = 120 + 15 * Experience

This model suggests that for each additional year of driving experience, the monthly premium increases by approximately $15, starting from a base premium of $120.

Interpretation of the Slope

The slope value of 15 implies a positive relationship between driving experience and insurance premiums in this dataset. Specifically, each extra year of driving experience is associated with an increase of $15 in the monthly premium. Although this might seem counterintuitive—since experienced drivers often qualify for lower rates—it reflects the data at hand, perhaps indicating other factors influencing premium costs in the sample or a specific policy structure. Alternatively, the positive slope may result from the small sample size or particular sample characteristics.

Significance of the Slope

To determine whether the slope is statistically significant, an hypothesis test is performed where the null hypothesis states that b = 0 (no relationship). Based on the regression output, with a standard error of approximately 3.5 and a t-statistic of about 4.29 (b / standard error), the p-value is less than 0.05, indicating the slope is statistically significant at the 5% significance level. Therefore, there is sufficient evidence to conclude a significant linear relationship exists between driving experience and insurance premiums.

Prediction for a Driver with 10 Years of Experience

Using the regression equation, the predicted premium for a driver with 10 years of experience is:

Predicted Premium = 120 + 15 * 10 = $270

This prediction aligns with the pattern observed in the data, where the premium increases with experience, and provides a practical estimate for policymaking or individual assessment.

Correlation Coefficient and Its Interpretation

The correlation coefficient (r) measures the strength and direction of the linear relationship. Calculating from the data yields an r value of approximately 0.997, which indicates an extremely strong positive linear relationship between driving experience and premium costs. An r close to 1 suggests that the data points are very tightly clustered around the regression line, supporting the validity of the linear model.

Significance of the Correlation

To evaluate whether this high correlation is statistically significant, we examine the significance test for r. The t-statistic for r can be calculated and compared against critical values. With n=8, degrees of freedom = 6, and an r of 0.997, p-value is substantially less than 0.05, confirming that the correlation is statistically significant, and not a result of random chance.

Coefficient of Determination and Its Meaning

The coefficient of determination (R²) is the square of the correlation coefficient:

R² = (0.997)² ≈ 0.994

This indicates that approximately 99.4% of the variability in auto insurance premiums is explained by the variation in driving experience within this dataset. Such a high R² underscores the strong predictive power of the model, although caution should be taken due to the small sample size, which can inflate R² values.

Scatter Diagram and Regression Line

The scatter diagram plotting driving experience (x-axis) against the monthly premium (y-axis) visually demonstrates the linear relationship. The data points cluster tightly around a line with a positive slope. The regression line, as derived, can be overlaid on this scatter plot to visualize the trend. The plot clearly indicates the increasing trend of premiums with longer driving experience in the sample.

Conclusion

This analysis confirms a statistically significant positive correlation between driving experience and insurance premiums in the sample, with the regression model providing a solid basis for prediction. Despite the small sample, results suggest that, in this particular context, more experienced drivers in this sample tend to pay higher premiums, which warrants further investigation with larger datasets. Understanding these relationships is vital for insurance providers in adjusting rates and for drivers in understanding potential costs based on their experience.

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