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Suppose that a regression line for some data transformed with logarithmspredicts that when y equals 8, log(%) will equal 1.603. What does theregression line predict y will equal when y equals 8? Round your answer to thenearest whole number.

Suppose That A Regression Line For Some Data Transformed With Logarithmspredicts That When Y Equals 8 Log Will Equal 1603 What Does Theregression Line Predict Y class=

Sagot :

Given the relationship between y and x to be

[tex]y=a^x\text{ ------ equation 1}[/tex]

Take the logarithm of both sides,

[tex]\begin{gathered} \log y=\log ^{}_{}a^x \\ \Rightarrow\log \text{ y = x }\times\text{ log a ---- equation 2} \end{gathered}[/tex]

But when x = 8, log y = 1.603.

Thus, substituting the above values into equation 2, we have

[tex]\begin{gathered} 1.603\text{ = 8 }\times\text{ log a} \\ \text{divide both sides by 8} \\ \log \text{ a= }\frac{1.603}{8} \\ \Rightarrow\log \text{ a =0.2}004 \\ \text{Thus, } \\ a=1.586 \end{gathered}[/tex]

From equation 1,

[tex]\begin{gathered} y=a^x \\ \Rightarrow y=1.586^x\text{ ----- equation 3} \end{gathered}[/tex]

Thus, when x = 8

[tex]\begin{gathered} y=1.586^x \\ y=1.586^8 \\ \Rightarrow y=40.03 \end{gathered}[/tex]

Thus, the value of y will be 40 (to the nearest whole number)

The correct option is D

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