cientists are studying the population of a particular type of fish. The table below shows the data gathered over a five-month time period. Use the data to answer questions 5-9.
number of months number of fish
0 8
1 39
2 195
3 960
4 4,738
5 23,375
5. what does the scatterplot of the data show?
1. strong positive linear relationship
2. a strong negative linear relationship
3. a curve that represents exponential growth
4. a curve that reprents ecponential decay
I select number 3
6. Complete an exponential transformation on the y-values. What is the new value of y when x+5?
1. 4.3688
2. 3.6756
3. 0.6990
4. 3.3757
I select #3
7. Find the linear transformation Model
1. log y hat=0.6935•logx + 0.9013
2. log y hat=0.9013x+0.6935
3. log y hat=0.6935x+0.9013
4.
log y hat=0.9013•logx+0.6935
8. Use the linear transformation model to predict the number of fish in 12 months.
this is where I need help.
9. A power model is shown below. Determine the residual for the observed data x=7 and y=70
log y hat=1.6+0.31log x
1.71.37
2.1.37
3.1.85
4.-1.37
I select 4
To answer question number 8, you need to use the linear transformation model that you found in question number 7. The linear transformation model is given as:
log y hat = 0.6935x + 0.9013
To predict the number of fish in 12 months, you need to substitute x = 12 into the equation and solve for log y hat:
log y hat = 0.6935 * 12 + 0.9013
Calculating this equation will give you the value of log y hat. To convert it back to the original scale, you need to exponentiate the value. This can be done by raising 10 to the power of log y hat:
y hat = 10^(log y hat)
Using the calculated value of log y hat, you can now find y hat, which represents the predicted number of fish in 12 months.
To answer question 8, we can use the linear transformation model to predict the number of fish in 12 months. The linear transformation model is given as:
log y hat = 0.6935x + 0.9013
To predict the number of fish in 12 months, we substitute x = 12 into the equation:
log y hat = 0.6935(12) + 0.9013
Now we can calculate this value:
log y hat = 8.322 + 0.9013
log y hat = 9.2233
To get the predicted value of y, we need to take the antilog of both sides of the equation:
y hat = 10^(9.2233)
y hat ≈ 1,780,840
Therefore, the predicted number of fish in 12 months is approximately 1,780,840.
For question 9, you have selected option 4 (-1.37) as the residual for the observed data x=7 and y=70. However, to calculate the residual, we need to use the power model equation provided:
log y hat = 1.6 + 0.31log x
To calculate the predicted value of y (y hat) for x = 7:
log y hat = 1.6 + 0.31log(7)
Now we can calculate this value:
log y hat = 1.6 + 0.31(0.8451)
log y hat = 1.6 + 0.2617
log y hat = 1.8617
To get the predicted value of y, we need to take the antilog of both sides of the equation:
y hat = 10^(1.8617)
y hat ≈ 70.7663
The observed value of y is given as 70. Therefore, the residual for this data point would be the difference between the observed value and the predicted value:
Residual = observed value - predicted value
= 70 - 70.7663
≈ -0.7663
So, the correct answer for question 9 is option 4 (-0.7663), not option 4 (-1.37).