Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Sure, let's address each part of the question step by step:
### (a) Calculate the differences, mean difference, and standard deviation for these differences
For each car, subtract the other garage's estimate from Jocko's estimate:
[tex]\[ \begin{aligned} & \text{Car 1: } 1410 - 1250 = 160 \\ & \text{Car 2: } 1550 - 1300 = 250 \\ & \text{Car 3: } 1250 - 1250 = 0 \\ & \text{Car 4: } 1300 - 1200 = 100 \\ & \text{Car 5: } 900 - 950 = -50 \\ & \text{Car 6: } 1520 - 1575 = -55 \\ & \text{Car 7: } 1750 - 1600 = 150 \\ & \text{Car 8: } 3600 - 3380 = 220 \\ & \text{Car 9: } 2250 - 2125 = 125 \\ & \text{Car 10: } 2840 - 2600 = 240 \\ \end{aligned} \][/tex]
The differences are: [tex]\([160, 250, 0, 100, -50, -55, 150, 220, 125, 240]\)[/tex]
Next, calculate the mean difference:
[tex]\[ \text{Mean Difference} = \frac{160 + 250 + 0 + 100 - 50 - 55 + 150 + 220 + 125 + 240}{10} = 114.0 \][/tex]
Calculate the standard deviation of the differences:
[tex]\[ \text{Standard Deviation} = 114.4018 \][/tex]
### (b) Test the null hypothesis that there is no difference between the estimates of the two garages
Null Hypothesis ([tex]\(H_0\)[/tex]): The mean difference in estimates between Jocko's Garage and the other garage is 0. ([tex]\(\mu_d = 0\)[/tex])
Alternative Hypothesis ([tex]\(H_a\)[/tex]): The mean difference in estimates between Jocko's Garage and the other garage is not 0. ([tex]\(\mu_d \neq 0\)[/tex])
To test this hypothesis, we use the paired sample t-test.
Calculate the t-statistic and p-value:
[tex]\[ \text{t-statistic} = 3.1512 \][/tex]
[tex]\[ \text{p-value} = 0.011715 \][/tex]
Since the p-value is less than 0.05, we reject the null hypothesis at the 5% significance level, indicating that there is a statistically significant difference in the estimates between the two garages.
### (c) Construct a 95% confidence interval for the difference in estimates
We construct a 95% confidence interval for the mean difference.
[tex]\[ \text{95% Confidence Interval} = (32.1619, 195.8381) \][/tex]
### (d) Recommend repayment for 1000 claims
Using the 95% confidence interval, we can recommend repayment for 1000 claims. To calculate this, we multiply the bounds of the confidence interval by the number of claims:
[tex]\[ \text{Lower Bound Repayment (for 1000 claims)} = 1000 \times 32.1619 = 32161.86 \text{ dollars} \][/tex]
[tex]\[ \text{Upper Bound Repayment (for 1000 claims)} = 1000 \times 195.8381 = 195838.14 \text{ dollars} \][/tex]
So, the insurance company could seek repayment ranging from approximately [tex]\(32162\)[/tex] dollars to [tex]\(195838\)[/tex] dollars for 1000 claims based on the 95% confidence interval.
### (a) Calculate the differences, mean difference, and standard deviation for these differences
For each car, subtract the other garage's estimate from Jocko's estimate:
[tex]\[ \begin{aligned} & \text{Car 1: } 1410 - 1250 = 160 \\ & \text{Car 2: } 1550 - 1300 = 250 \\ & \text{Car 3: } 1250 - 1250 = 0 \\ & \text{Car 4: } 1300 - 1200 = 100 \\ & \text{Car 5: } 900 - 950 = -50 \\ & \text{Car 6: } 1520 - 1575 = -55 \\ & \text{Car 7: } 1750 - 1600 = 150 \\ & \text{Car 8: } 3600 - 3380 = 220 \\ & \text{Car 9: } 2250 - 2125 = 125 \\ & \text{Car 10: } 2840 - 2600 = 240 \\ \end{aligned} \][/tex]
The differences are: [tex]\([160, 250, 0, 100, -50, -55, 150, 220, 125, 240]\)[/tex]
Next, calculate the mean difference:
[tex]\[ \text{Mean Difference} = \frac{160 + 250 + 0 + 100 - 50 - 55 + 150 + 220 + 125 + 240}{10} = 114.0 \][/tex]
Calculate the standard deviation of the differences:
[tex]\[ \text{Standard Deviation} = 114.4018 \][/tex]
### (b) Test the null hypothesis that there is no difference between the estimates of the two garages
Null Hypothesis ([tex]\(H_0\)[/tex]): The mean difference in estimates between Jocko's Garage and the other garage is 0. ([tex]\(\mu_d = 0\)[/tex])
Alternative Hypothesis ([tex]\(H_a\)[/tex]): The mean difference in estimates between Jocko's Garage and the other garage is not 0. ([tex]\(\mu_d \neq 0\)[/tex])
To test this hypothesis, we use the paired sample t-test.
Calculate the t-statistic and p-value:
[tex]\[ \text{t-statistic} = 3.1512 \][/tex]
[tex]\[ \text{p-value} = 0.011715 \][/tex]
Since the p-value is less than 0.05, we reject the null hypothesis at the 5% significance level, indicating that there is a statistically significant difference in the estimates between the two garages.
### (c) Construct a 95% confidence interval for the difference in estimates
We construct a 95% confidence interval for the mean difference.
[tex]\[ \text{95% Confidence Interval} = (32.1619, 195.8381) \][/tex]
### (d) Recommend repayment for 1000 claims
Using the 95% confidence interval, we can recommend repayment for 1000 claims. To calculate this, we multiply the bounds of the confidence interval by the number of claims:
[tex]\[ \text{Lower Bound Repayment (for 1000 claims)} = 1000 \times 32.1619 = 32161.86 \text{ dollars} \][/tex]
[tex]\[ \text{Upper Bound Repayment (for 1000 claims)} = 1000 \times 195.8381 = 195838.14 \text{ dollars} \][/tex]
So, the insurance company could seek repayment ranging from approximately [tex]\(32162\)[/tex] dollars to [tex]\(195838\)[/tex] dollars for 1000 claims based on the 95% confidence interval.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
