Question

A biologist studying spotted 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. \[\widehat{\log(\text{weight})}=3.11\log(\text{length})-4.94\] according to this model, what is the predicted weight of a spotted bass that is \[50\,\text{cm}\] long? you may round your answer to one decimal place.

Answer 1

a) According to the given least-squares regression equation, the predicted weight of a spotted bass that is 50 cm long is approximately 2.2 kg.

Step-by-step explanation:

The given least-squares regression equation is
`\(\widehat{\log(\text{weight})} = 3.11 \cdot \log(\text{length}) - 4.94\).` To find the predicted weight for a spotted bass of 50 cm length, we substitute
`\( \log(\text{length}) = \log(50) \)` into the equation.

`\[ \widehat{\log(\text{weight})} = 3.11 \cdot \log(50) - 4.94 \]`

Now, calculate the expression:

`\[ \widehat{\log(\text{weight})} = 3.11 \cdot 1.69897 - 4.94 \]`

`\[ \widehat{\log(\text{weight})} \approx 0.34257 \]`

To find the predicted weight, we need to undo the logarithm transformation. So, we calculate
`\( \text{weight} = 10^(0.34257) \).`

`\[ \text{weight} \approx 2.18 \, \text{kg} \]`

Rounding to one decimal place, the predicted weight of a spotted bass that is 50 cm long is approximately 2.2 kg. The logarithmic transformation helps linearize the relationship between length and weight, making it easier to model and interpret. The regression equation provides a tool to estimate the weight of a spotted bass based on its length, allowing for more efficient and accurate biological studies.