An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Perform linear, quadratic, and exponential regressions on the given data.
Compare the R 2 values of the three models to determine the best fit.
Choose the quadratic model ( C = 2.36 t 2 + 24.16 t + 647.00 ) as the best fit due to its highest R 2 value ( 0.9941 ).
Predict the average spending in 2025 by substituting t = 23 into the quadratic equation, resulting in 2451.12 .
Explanation
Data Analysis and Model Selection First, let's analyze the data provided. We have the average consumer spending on Christmas shopping for the years 2002 to 2007. We need to find the best-fitting mathematical model (linear, quadratic, or exponential) and then use it to predict the spending in 2025.
Regression Analysis We have performed linear, quadratic, and exponential regressions on the data. The results are as follows:
Linear Regression: C = 35.94 t + 639.14 , R 2 = 0.9850
Quadratic Regression: C = 2.36 t 2 + 24.16 t + 647.00 , R 2 = 0.9941
Exponential Regression: C = 642.39 × exp ( 0.05 t ) , R 2 = 0.9904
The R 2 value represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). A higher R 2 value indicates a better fit.
Model Comparison Comparing the R 2 values, we see that the quadratic regression has the highest value ( 0.9941 ), followed by the exponential regression ( 0.9904 ), and then the linear regression ( 0.9850 ). Therefore, the quadratic model provides the best fit for the given data.
Justification of Model Choice The quadratic model is the best choice because it has the highest R 2 value, indicating that it explains the most variance in the data compared to the linear and exponential models. While the exponential model also has a high R 2 value, the quadratic model is slightly better. The linear model has the lowest R 2 value, indicating the poorest fit.
Prediction for 2025 Now, we need to predict the average spending in 2025. First, we calculate the value of t for 2025:
t = 2025 − 2002 = 23
Now, we substitute t = 23 into the quadratic regression equation:
C = 2.36 ( 23 ) 2 + 24.16 ( 23 ) + 647.00
C = 2.36 ( 529 ) + 555.68 + 647.00
C = 1248.44 + 555.68 + 647.00
C = 2451.12
Final Prediction Therefore, based on the quadratic regression model, the predicted average amount of money spent by consumers on their Christmas shopping in 2025 is approximately $2451.12.
Examples
Understanding consumer spending habits is crucial for businesses and economists alike. By analyzing historical data and creating predictive models, businesses can forecast future sales and adjust their inventory and marketing strategies accordingly. For example, a retailer might use this type of analysis to determine how much inventory to stock for the holiday season, or a marketing team might use it to optimize their advertising campaigns. Economists can use these models to understand broader economic trends and make predictions about future economic growth.
