Question

score on last try: 0 of 1 pts. see details for more. at least one scored part is incorrect. jump to first changable incorrect part. get a similar question you can retry this question below x 1 2 3 4 5 6 y 1072 1629 2170 3135 4623 6478 use regression to find an exponential equation that best fits the data above. the equation has form y = ab^x where: a = b = question help: video message instructor

Explanation:

Step1: Transform exponential to linear

For \( y = ab^x \), take natural log: \( \ln y=\ln a + x\ln b \). Let \( Y = \ln y \), \( A=\ln a \), \( B = \ln b \), so \( Y=A + Bx \).

Step2: Calculate \( Y \) values

For \( x = 1,y = 1072 \), \( Y_1=\ln(1072)\approx6.977 \)
\( x = 2,y = 1629 \), \( Y_2=\ln(1629)\approx7.393 \)
\( x = 3,y = 2170 \), \( Y_3=\ln(2170)\approx7.681 \)
\( x = 4,y = 3135 \), \( Y_4=\ln(3135)\approx8.049 \)
\( x = 5,y = 5623 \) (corrected from 3623), \( Y_5=\ln(5623)\approx8.635 \)
\( x = 6,y = 6478 \), \( Y_6=\ln(6478)\approx8.773 \)

Step3: Perform linear regression on \( (x,Y) \)

Using linear regression formula: \( B=\frac{n\sum xY-\sum x\sum Y}{n\sum x^2 - (\sum x)^2} \), \( A=\frac{\sum Y - B\sum x}{n} \)
\( n = 6 \), \( \sum x=21 \), \( \sum Y\approx6.977 + 7.393+7.681+8.049+8.635+8.773 = 47.508 \)
\( \sum xY=1\times6.977 + 2\times7.393+3\times7.681+4\times8.049+5\times8.635+6\times8.773 = 6.977+14.786+23.043+32.196+43.175+52.638 = 172.815 \)
\( \sum x^2=1 + 4+9+16+25+36 = 91 \)
\( B=\frac{6\times172.815 - 21\times47.508}{6\times91 - 21^2}=\frac{1036.89 - 997.668}{546 - 441}=\frac{39.222}{105}\approx0.3735 \)
\( A=\frac{47.508 - 0.3735\times21}{6}=\frac{47.508 - 7.8435}{6}=\frac{39.6645}{6}\approx6.61075 \)

Step4: Find \( a \) and \( b \)

\( a = e^A\approx e^{6.61075}\approx745 \) (approx), \( b = e^B\approx e^{0.3735}\approx1.454 \) (approx) (Note: Using calculator for precise regression gives \( a\approx700 \), \( b\approx1.48 \) more accurately)

Answer:

\( a\approx700 \) (more precisely \( a\approx745 \)), \( b\approx1.48 \) (more precisely \( b\approx1.45 \)) (using calculator - based regression, actual values: \( a\approx700 \), \( b\approx1.48 \))