QuestionJune 21, 2025

The Consumer Price Index (CPI) is a measure of the change in the cost of goods over time. The index was 100 for the three-year period centered on 1983. For simplicity. we will assume that the CPI was exactly 100 in 1982. Then the CPI of 1936 in 2006 indicates that an item that cost 1.00 in 1982 would cost 1.94 in 2006. The CPI has been increasing approximately linearly over the last few decades a. Use this information to determine an equation for the CPI in terms of 1, which represents the years since 1982 b. Based on the answer to part (a)what was the predicted value of the CPI in 2000? Compare this estimate with the actual CPI of 169 c. Describe the rate at which the annual CPI is changing

The Consumer Price Index (CPI) is a measure of the change in the cost of goods over time. The index was 100 for the three-year period centered on 1983. For simplicity. we will assume that the CPI was exactly 100 in 1982. Then the CPI of 1936 in 2006 indicates that an item that cost 1.00 in 1982 would cost 1.94 in 2006. The CPI has been increasing approximately linearly over the last few decades a. Use this information to determine an equation for the CPI in terms of 1, which represents the years since 1982 b. Based on the answer to part (a)what was the predicted value of the CPI in 2000? Compare this estimate with the actual CPI of 169 c. Describe the rate at which the annual CPI is changing
The Consumer Price Index (CPI) is a measure of the change in the cost of goods over time. The index was 100 for the three-year period centered on 1983. For simplicity. we will assume that the CPI was exactly 100 in 1982. Then the CPI of 1936 in 2006 indicates that an item that cost 1.00 in 1982 would cost 1.94 in 2006. The CPI has been increasing approximately linearly over the last few decades a. Use this information to determine an equation for the CPI in terms of 1, which represents the years since 1982 b. Based on the answer to part (a)what was the predicted value of the CPI in 2000? Compare this estimate with the actual CPI of 169 c. Describe the rate at which the annual CPI is changing

Solution
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Answer

a. CPI(t) = 3.9t + 100 ### b. Predicted CPI in 2000: 170.2; Actual CPI: 169 ### c. Annual CPI change rate: 3.9 units/year Explanation 1. Determine the slope of CPI increase The slope m is calculated using **m = \frac{y_2 - y_1}{x_2 - x_1}**. Here, y_2 = 193.6, y_1 = 100, x_2 = 2006 - 1982 = 24, x_1 = 0. So, m = \frac{193.6 - 100}{24} = 3.9. 2. Formulate the equation for CPI Use the point-slope form **y = mx + b**. With b = 100 (CPI in 1982), the equation becomes CPI(t) = 3.9t + 100. 3. Calculate predicted CPI for 2000 Substitute t = 2000 - 1982 = 18 into CPI(t) = 3.9t + 100: CPI(18) = 3.9 \times 18 + 100 = 170.2. 4. Compare predicted CPI with actual CPI in 2000 Predicted CPI is 170.2; actual CPI is 169. The prediction is slightly higher by 1.2. 5. Describe the rate of change of CPI The annual rate of change is the slope m = 3.9. This means the CPI increases by 3.9 units per year.

Explanation

1. Determine the slope of CPI increase<br /> The slope $m$ is calculated using **$m = \frac{y_2 - y_1}{x_2 - x_1}$**. Here, $y_2 = 193.6$, $y_1 = 100$, $x_2 = 2006 - 1982 = 24$, $x_1 = 0$. So, $m = \frac{193.6 - 100}{24} = 3.9$.<br /><br />2. Formulate the equation for CPI<br /> Use the point-slope form **$y = mx + b$**. With $b = 100$ (CPI in 1982), the equation becomes $CPI(t) = 3.9t + 100$.<br /><br />3. Calculate predicted CPI for 2000<br /> Substitute $t = 2000 - 1982 = 18$ into $CPI(t) = 3.9t + 100$: $CPI(18) = 3.9 \times 18 + 100 = 170.2$.<br /><br />4. Compare predicted CPI with actual CPI in 2000<br /> Predicted CPI is 170.2; actual CPI is 169. The prediction is slightly higher by 1.2.<br /><br />5. Describe the rate of change of CPI<br /> The annual rate of change is the slope $m = 3.9$. This means the CPI increases by 3.9 units per year.
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