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The assignment involves analyzing a dataset related to collision coverage claims collected by the Highway Loss Data Institute for 2007 models. The specific tasks include calculating and interpreting the first, second (median), and third quartiles for the provided collision coverage claims data. Additionally, the assignment prompts interpretation of the quartile results in the context of insurance claims.

Furthermore, the assignment encompasses conducting statistical tests for independence between variables in two different contexts: marital status and happiness, and smoking status and education level. For each case, you are required to complete expected counts, perform a chi-square test at a 95% confidence level, and interpret your findings. Bar charts should be constructed to visualize the distribution of smoking status by years of education, and conclusions should address whether the evidence supports the independence of the variables tested.

Paper For Above instruction

The analysis of collision coverage claims provides insights into the central tendency and distribution of damages reported in the vehicle insurance sector. The dataset includes several claims ranging from modest amounts to substantial losses, and understanding the quartiles of this data helps insurers and policymakers evaluate the spread, identify typical claim sizes, and assess risk levels associated with vehicle damage claims. Furthermore, applying statistical tests to examine relationships between different social variables—such as marital status with happiness, and education level with smoking status—enables researchers to identify potential associations or independence between these categorical variables, informing policy decisions and social interventions.

Quartile Calculations and Interpretation

The dataset for collision claims is as follows: $6,751; $9,908; $3,461; $21,147; $2,332; $2,336; $189; $1,185; $370; $1,414; $4,668; $1,953; $10,034; $735; $802; $618; $180; $1,657. To determine the quartiles, the data should first be ordered from smallest to largest:

• $180
• $189
• $370
• $618
• $735
• $802
• $1,185
• $1,414
• $1,953
• $2,332
• $2,336
• $3,461
• $6,751
• $9,908
• $10,034
• $21,147

The ordered dataset contains 16 values. The second quartile (median) is the average of the 8th and 9th values:

• 8th value: $1,414
• 9th value: $1,953

Median (Q2) = ($1,414 + $1,953) / 2 = $1,683.50

The first quartile (Q1) is the median of the lower half (first 8 data points):

• Lower half data: $180, $189, $370, $618, $735, $802, $1,185, $1,414
• Q1 = median of these 8 values = average of 4th and 5th: ($618 + $735)/2 = $676.50

The third quartile (Q3) is the median of the upper half (last 8 data points):

• Upper half data: $1,953, $2,332, $2,336, $3,461, $6,751, $9,908, $10,034, $21,147
• Q3 = median of these 8 values = average of 4th and 5th: ($3,461 + $6,751)/2 = $5,106

Interpretively, the first quartile ($676.50) indicates that 25% of collision claims are below this value, reflecting relatively minor damages; the median ($1,683.50) suggests that half of the claims are below this amount, while the third quartile ($5,106) indicates that 75% of the claims are below this higher threshold. These quartiles inform insurers about the typical and extreme claim amounts, aiding in risk assessment and premium setting.

Analysis of Marital Status and Happiness

The second part of the assignment examines whether there is an association between marital status and happiness levels. The data structure includes categories for marital status (Married, Widowed, Divorced/Separated, Never Married) and happiness levels (Very happy, Pretty happy, Not too happy). Completing the expected counts involves calculating expected frequencies for each cell based on the marginal totals using the formula:

Expected Count = (row total * column total) / grand total

For example, if the total number of married individuals is R1, the total number of very happy individuals is C1, and the overall total is N, then expected count for married & very happy = (R1 * C1)/N.

Once the expected counts are calculated, the chi-square statistic is computed by summing [(observed - expected)^2 / expected] across all cells. The degrees of freedom are determined by (rows - 1) * (columns - 1). Comparing the chi-square statistic with the critical value for df at 0.05 significance level reveals whether the variables are independent.

The initial calculation indicates that if the chi-square value exceeds the critical value, we reject the null hypothesis of independence, implying an association exists between marital status and happiness. Conversely, if it is less, we conclude no significant relationship exists, supporting the hypothesis of independence.

Analysis of Smoking Status and Education Level

Next, the data for smoking status and education level is analyzed. Expected counts are calculated similarly based on marginal totals of the categories like 16 years of education combined with the categories Current, Former, Never for smoking status.

Constructing a bar chart involves plotting the counts of each smoking status against the levels of education, providing a visual representation of the data distribution. The bar chart facilitates quick assessment of whether smoking prevalence varies with education level.

Performing the chi-square test as before determines whether the relationship between smoking status and education level is statistically significant at the 95% confidence level. If the p-value obtained from the chi-square statistic is less than 0.05, we conclude that smoking status and education level are not independent; otherwise, they are likely independent.

The results from these tests contribute valuable insights into health behaviors and social determinants, potentially informing public health policies aimed at reducing smoking rates among various educational groups.

Conclusions

The statistical analyses indicate the levels of claims within the insurance dataset and reveal potential associations between social variables such as marital status, happiness, smoking, and education. The quartile computations shed light on the typical and extreme claim amounts, guiding risk management strategies. The chi-square tests provide evidence for or against the independence of social variables, which can influence targeted interventions or policy adjustments. Ultimately, such thorough statistical evaluations enhance understanding of human behavior and risk factors, vital for insurers, policymakers, and social scientists.

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