Question

jean invested $380 in stocks. over the next 5 years, the value of her investment grew, as shown in the accompanying table.
years since investment (x) | value of stock, in dollars (y)
0 | 380
1 | 395
2 | 411
3 | 427
4 | 445
5 | 462
write the exponential regression equation for this set of data, rounding all values to two decimal places. using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase.
value of stock =

Explanation:

Step1: Recall Exponential Regression Formula

The general form of an exponential regression equation is \( y = ab^x \), where \( a \) and \( b \) are constants to be determined. To find \( a \) and \( b \), we can use a calculator or software (like a TI - 84 Plus or Excel) with exponential regression capabilities.

First, we list the data points: \((0, 380)\), \((1, 395)\), \((2, 411)\), \((3, 427)\), \((4, 445)\), \((5, 462)\)

Step2: Calculate Exponential Regression

Using a graphing calculator or statistical software:

• For a TI - 84 Plus:
• Enter the data into lists. Let \( L_1 \) be the \( x \)-values (0, 1, 2, 3, 4, 5) and \( L_2 \) be the \( y \)-values (380, 395, 411, 427, 445, 462).
• Press STAT, then CALC, then ExpReg (exponential regression).
• The calculator will give \( a\approx380.00 \) (since when \( x = 0 \), \( y=a b^0=a\), and the initial value is 380) and \( b\approx1.04 \) (after rounding to two decimal places). So the exponential regression equation is \( y = 380.00\times(1.04)^x \)

Step3: Predict the Value at \( x = 10 \)

Now, we want to find the value of \( y \) when \( x = 10 \). Substitute \( x = 10 \) into the equation \( y=380.00\times(1.04)^x \)

First, calculate \( (1.04)^{10} \). Using a calculator, \( (1.04)^{10}\approx1.480244 \)

Then, multiply by 380: \( y = 380.00\times1.480244\approx562.49 \)

Rounding to the nearest dollar, we get \( y\approx562 \)

Answer:

The value of her stock 10 years after her initial purchase is \(\boxed{562}\)