Insurance Rates By Company And Essay, Research Paper
The problem under investigation is to determine whether or not there is a statistically significant difference in the insurance rates by company and driver type. Data on the insurance rates were obtained from the Missouri Department of Insurance. The two-way analysis of variance procedure was used to reach valid conclusions. The study concluded that there is actually no difference in insurance rates among companies, but a difference in insurance rates among driver types. Therefore, there is no evidence that companies differ significantly in their insurance rates by companies, but there is evidence that companies differ significantly in their insurance rates by driver type.
Insurance rates are no doubt very critical to both the companies and drivers. Missouri requires you to have a minimum amount of auto insurance coverage s. One is a bodily injury liability of 25/50 (up to $25,000 for injury-related expenses incurred by a single individual in an accident that is your fault; up to $50,000 for all such expenses from a single accident.) Another is property damage liability of up to $10,000. The other is uninsured motorist protection of 25/50.
The purpose of this paper is to use valid statistical procedures to determine whether or not the four insurance companies in Jefferson City, Missouri differed significantly in their insurance rates for the period of six months in the year of 1996. Extensive search was made on the Internet to locate relevant and reliable data on insurance rates. The homepage of the Insurance guide (www.insure.com) provided me with the required data on insurance rates in Jefferson City, Missouri. The data is shown in Appendix A.
The data in Appendix A is arranged in a format of a randomized block design (RCBD) where blocks is the insurance companies and the primary factor of interest are the single or married/sex/age. The statistical procedure that I used is the two-way analysis of variance (ANOVA) without replication. The Microsoft Excel software package was used to perform all necessary computations and analyses.
Bases on the analyses performed in this paper, I reached the conclusion that actually, there ware significant differences in the insurance rates of the four insurance companies for the period of six months in 1996.
METHODS AND PROCEDURES USED
A search through the Internet was conducted to locate data sources for insurance rates in the U.S. For accuracy and reliability reasons, we decided to obtain our data from the homepage of the Insurance guide (www.insure.com). Table A-1, A-2, and A-3 in the Appendix, gives the insurance rates of four insurance companies in Jefferson City, Missouri for the year of 1996, respectively. This guide includes rate comparisons using six-month premiums based on rates in effect as of July 1, 1996. In addition to specifics explained later under “Types of Drivers,” the rates assume $1,000 in medical payments, that the driver travels 10 miles or less to work each day and drives approximately 15,000 miles a year. Preferred drivers, standard drivers, and non-standard drivers are the types of drivers used in the Missouri Auto Rate Guides. These examples use four hypothetical drivers: 19-year old male, 19 year-old female, 40 year-old adults, and 60 year-old adults. Gender-based rates tend to disappear after age 30. Preferred drivers get the best rates and have no accidents or moving violations on their recent record. In these examples, they carry liability coverage s of $100,000/300,000/50,000 (compared with Missouri minimums of $25,000/50,000/10,000). They have comprehensive and collision coverage’s with a $250 deductible. They drive a 1991 Ford Taurus. Standard drivers get average rates and, in these examples, have two moving violations on their record, carry liability coverage of $50,000/100,000/50,000 and comprehensive and collision with a $250 deductible. They drive a 1991 Honda Accord EX. Non-standard drivers have poor driving records and may have an at-fault accident and two speeding tickets on their record. They carry the minimum required liability coverage ($25,000/$50,000/$10,000) and no collision or comprehensive.
The data in Tables A-1, A-2, and A-3 were reorganized to form an RCBD structure as shown in Table B-1, B-2, and B-3. The RCBD in Table A-4 through A-6 contains four rows and four columns. The rows are the four groups of single male, 19/single female, 19/married male, 62/married female, 40 and they represent the levels of the primary factor under study, which is the driver type factor. The columns are the four companies and they represent the secondary factor (insurance companies), which is the company factor. The RCBD was chosen as the appropriate experimental design in our investigation because it allows us to account for and remove the influence of the blocks (insurance companies) that may affect comparisons among the levels pf the primary factor (the single or married/sex/age).
The analysis of an RCBD requires the execution of several steps (see, e.g., Introduction To Business Statistics, Kvanli, et al 2000, pp. 475-481).
The first step in the analysis requires writing a null hypothesis and an alternative hypothesis for each of the primary factor (company) and for the block factor (driver type) as follow:
To determine whether or not there is significant difference in the insurance rates among the four insurance companies, we test the hypotheses.
Null hypothesis: H There is no difference in insurance rates among companies, (1)
Alternative hypothesis: H There is difference in insurance rates among companies.
To determine whether or not there is significant difference in the insurance rates among driver type, we test the hypothesizes:
Null hypothesis: H There is no difference in insurance rates among driver type, (2)
Alternative hypothesis: H There is difference in insurance rates among driver type.
The second step in the analysis requires the calculation of the primary factor (company) totals (i.e., the row totals: T , T , , T ), and the block (driver type totals (i.e., the columns totals: B , B , B ). Also we need to calculate the grand total:
The third step in the analysis requires the calculation of the following sums of squares (SS): SS(total), SS(factor), SS(block), and SS(error).
SS(total) measures the total variation present in the entire set of data and it is computed using the formula:
SS(factor) measures the amount of variation due to differences among the levels of the primary factor (the companies) and it is computed from:
SS(blocks) measures the amount of variation due to differences among blocks (the driver types) and it is computed from:
SS(error) measures the amount of variation due to all other sources that were not accounted for in the RCBD. SS(error) is computed from:
SS(error) = SS(total) – SS(factor) – SS(blocks). (7)
The fourth step in the analysis requires the calculation of the following mean squares (MS): MS(factor), MS(blocks), and MS(error) according to the following formulas:
MS(factor) = SS(factor)/(k-1), (8)
MS(blocks) = SS(factor)/(b-1), (9)
MS(error) = SS(error)/(b-l)(k-l). (10)
The fifth step in the analysis requires the calculation of one F-ratio statistic for testing for the factor effect and on another F-ratio statistic for testing for the block effect. The formulas for these F-ratio statistics are:
F(factor) = MS(factor)/MS(error). (l1)
F(factor) has an F-distribution with (k-1, (b-1)(k-1)) degrees of freedom when the null hypothesis in (1) is true.
F(blocks) = MS(blocks)/MS(error). (12)
F(blocks) has an F-distribution with (b-1,(b-1) (k-1)) degrees of freedom when the null hypothesis in (2) is true.
The sixth step in the analysis requires the determination of decision rules for the rejection (or non-rejection) of the null hypotheses in (1) and (2). Usually, one of two approaches may be followed.
Using a pre-assigned level of significance , a rejection region is determined by reading statistical tables (the F-distribution tables in the case of our investigation) to obtain a critical value for the rejection region. Table A.7 in Kvanli et al (2000) provides the critical values for the F-distribution. In this paper a level of significance of 0.05 was used for all testing procedures. In general, the significance level of a statistical test procedure is the chance (that the investigator is willing to tolerate) of falsely rejecting a giving null hypothesis. Actually the critical value is designated as F crit .
Without the need to refer to statistical tables (the F-tables in this case), we just need to read the P-value of the test as printed in the computer output. In general, the P-value of a certain test procedure is the smallest significance level at which the given null hypothesis can be rejected. The smaller the P-value, the more we are inclined to reject the null hypothesis. Kvanli, et al (2000, p. 329) stated the following rule.
P-Value Rule of Thumb:
The test procedure rejects the null hypothesis if the P-value is less than 0.01, the test procedure fails to reject the null hypothesis if the P-value is greater than 0.10, the test procedure is inconclusive if the P-value falls between 0.01 and 0.10, inclusive. Microsoft Excel provides a P-value in all its test procedures.
All calculations required in our investigation were executed using a personal computer. The Microsoft Excel procedure ANOVA: Two-Factor without Replication was invoked to perform all the necessary calculations such as the sums of squares, the mean squares, and the F-rations in eq. (3)- (12). Results of the calculations are displayed in Table 1, which is called the two-way ANOVA table. Table 1 shows the sources of variation, the sums of squares (SS), the degrees of freedom, the mean squares (MS), the computed F-ratios (F), the P-values, and the critical F values (F crit).
ANOVA Table for the Randomized Complete Block Design
ANOVA Table C-1
Source of Variation SS df MS F P-value F crit
Rows (blocks) 1324273.688 3 441424.5625 108.1906374 2.2267E-07 3.86253873
Columns (factors) 137625.6875 3 45875.22917 11.24375648 0.00212503 3.86253873
Error 36720.5625 9 4080.0625
Total 1498619.938 15
RESULTS AND CONCLUSIONS
Testing for the primary factor (company) effect
The hypotheses to be tested are:
Null hypothesis Ho: There is no difference in insurance rates among companies,
Alternative hypothesis Ha: There is a difference in insurance rates among companies.
From Table 1, it is seen that the F-ratio and the P-value for testing for the primary effect (companies) are F-ratio= 108.190637398324 with P-value= 2.22674252183177. Based on the P-value and in accordance with the rule of thumb for the P-value, the conclusion is to fail to reject the null hypothesis. Therefore there is no evidence that companies differ significantly in their insurance rates by companies.
Testing for the block (driver type) effect
The hypotheses to be tested are:
Null hypothesis Ho: There is no difference in insurance rates among driver types,
Alternative hypothesis Ha: There is a difference in insurance rates among driver types.
From Table 1, it is seen that the F-ratio and the P-value for testing for the blocks effect (driver type) are F-ratio= 11.2437564784036 with P-value= 0.00212503454518515. Based on the P-value and in accordance with the rule of thumb for the P-value, the conclusion is to reject the null hypothesis. Therefore there is evidence that companies differ significantly in their insurance rates by driver type.
Further Investigation was recommended for other drivers wanted. The hypotheses tested for the primary factor (company) effect for the other tables were the same outcome. Also the hypotheses tested for the block (driver type) effect for the other tables were the same.
1. Introduction To Business Statistics: A computer integrated approach , Kvanli- 5th Edition, copyright 2000 by South-Western College Publishing
2. The Insurance Guide , www.insure.com, News Network, LLC, copyright 1195-2001