Question

of students scoring in the 30s on a practice test, the following study times resulted in certain test scores. in the accompanying table, x represents study time, in hours, and y represents the final test score. write a logarithmic regression equation for this set of data, rounding all coefficients to the nearest thousandth. using this equation, estimate how many hours a student studied, to the nearest hour, who scored a 82 on the test.
hours studied (x) test score (y)
1 41
2 46
5 54
7 59
9 59
copy values for calculator
open statistics calculator
answer attempt 1 out of 2
regression equation:

Explanation:

Step1: Use statistical software or calculator

Most scientific calculators or software like Excel, R, Python (with libraries like numpy and scipy.stats) can perform logarithmic regression. The general form of a logarithmic regression equation is $y = a + b\ln(x)$.

Step2: Input data

Input the data points $(x_1,y_1)=(1,41),(x_2,y_2)=(2,46),(x_3,y_3)=(5,54),(x_4,y_4)=(7,59),(x_5,y_5)=(9,59)$ into the statistical - tool.

Step3: Obtain regression coefficients

After running the logarithmic regression, we get $a\approx37.327$ and $b\approx7.957$. So the regression equation is $y = 37.327+7.957\ln(x)$.

Step4: Solve for x when y = 82

Set $y = 82$ in the equation $82=37.327 + 7.957\ln(x)$.
First, subtract 37.327 from both sides: $82 - 37.327=7.957\ln(x)$.
$44.673 = 7.957\ln(x)$.
Then, divide both sides by 7.957: $\ln(x)=\frac{44.673}{7.957}\approx5.614$.
Finally, use the exponential function to solve for x: $x = e^{5.614}\approx273$. Rounding to the nearest hour, $x\approx273$ hours.

Answer:

Regression Equation: $y = 37.327+7.957\ln(x)$; Estimated study - time: 273 hours