Question

A biologist studying florida bass observed a quadratic relationship between the lengths and weights of fish in a large sample. the biologist took the base \[10\] logarithm for the values of both variables, and they noticed a linear relationship in the transformed data. here's the least-squares regression equation for the transformed data, where weight is in kilograms and length is in centimeters. \[\widehat{\log(\text{weight})}=3.08\log(\text{length})-5.07\] according to this model, what is the predicted weight of a florida bass that is \[70\,\text{cm}\] long? you may round your answer to one decimal place. \[\text{kg}\]

Answer 1

To predict the weight of a Florida bass that is 70 cm long, we calculate the base 10 logarithm of the length, plug it into the regression equation, and then convert the logarithmic prediction back to a weight value. The estimated weight is found to be approximately 4.7 kg.

Step-by-step explanation:

To predict the weight of a Florida bass that is 70 cm long using the given least-squares regression equation for the transformed data, we need to substitute the logarithm of the length into the equation and solve for the weight. The regression equation is:

(\widehat{\log(\text{weight})}=3.08\log(\text{length})-5.07\)

First, we find the base 10 logarithm of the length:

(\log(70) \approx 1.8451\)

Now, we can use this value in the regression equation:

(\widehat{\log(\text{weight})} = 3.08 \cdot 1.8451 - 5.07 \approx 0.6695\)

Then, to find the actual predicted weight, we raise 10 to the power of the predicted logarithm:

(\text{weight} = 10^{\widehat{\log(\text{weight})}} = 10^{0.6695} \approx 4.7\,\text{kg}\)

Therefore, the predicted weight of a Florida bass that is 70 cm long is approximately 4.7 kg.