Let's walk through the steps to solve the question.
### Part (a):
First, we need to find the natural logarithm ([tex]\(\ln t\)[/tex]) for each value of [tex]\(t\)[/tex] in the table and then perform a linear regression to estimate the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the model [tex]\(P = a \ln t + b\)[/tex].
Given [tex]\(t\)[/tex] values (in minutes):
[tex]\[ 5, 15, 30, 60, 120, 240, 480, 720, 1440, 2880, 5760, 10080 \][/tex]
And the corresponding [tex]\(P\)[/tex] values (percentages):
[tex]\[ 73, 61.7, 58.3, 55.7, 50.3, 46.7, 40.3, 38.3, 29, 24, 18.7, 10.3 \][/tex]
Using linear regression, the estimated values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are calculated as:
[tex]\[ a = -7.7867 \][/tex]
[tex]\[ b = 86.2831 \][/tex]
These values are rounded to 4 decimal places.
### Part (b):
To predict when the subjects will recognize no words ([tex]\(P = 0\)[/tex]), we solve for [tex]\(t\)[/tex] in the equation [tex]\(0 = a \ln t + b\)[/tex].
[tex]\[
0 = -7.7867 \ln t + 86.2831
\][/tex]
Rearrange the equation to solve for [tex]\(\ln t\)[/tex]:
[tex]\[
\ln t = \frac{86.2831}{7.7867}
\][/tex]
[tex]\[
\ln t \approx 11.083
\][/tex]
Now, exponentiate both sides to solve for [tex]\(t\)[/tex]:
[tex]\[
t = e^{11.083} \approx 64914.4321 \text{ minutes}
\][/tex]
Convert minutes to days:
[tex]\[
\text{Days} = \frac{64914.4321}{1440} \approx 45.0797
\][/tex]
Thus:
[tex]\[ t \approx 45.0797 \text{ days} \][/tex]
### Part (c):
To predict when the subjects will recognize all words ([tex]\(P = 100\)[/tex]), we solve for [tex]\(t\)[/tex] in the equation [tex]\(100 = a \ln t + b\)[/tex].
[tex]\[
100 = -7.7867 \ln t + 86.2831
\][/tex]
Rearrange the equation to solve for [tex]\(\ln t\)[/tex]:
[tex]\[
\ln t = \frac{100 - 86.2831}{-7.7867}
\][/tex]
[tex]\[
\ln t \approx -1.764
\][/tex]
Now, exponentiate both sides to solve for [tex]\(t\)[/tex]:
[tex]\[
t = e^{-1.764} \approx 0.1718 \text{ minutes}
\][/tex]
Convert minutes to seconds:
[tex]\[
t \times 60 \approx 0.1718 \times 60 \approx 10.3064 \text{ seconds}
\][/tex]
Thus:
[tex]\[ t \approx 10.3064 \text{ seconds} \][/tex]
### Summary:
(a)
- [tex]\( a = -7.7867 \)[/tex]
- [tex]\( b = 86.2831 \)[/tex]
(b) Subjects will recognize no words in approximately 45.0797 days.
(c) Subjects will recognize all words in approximately 10.3064 seconds.
