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 8, we need to use the linear transformation model. From the options provided, the correct linear transformation model is:
log y hat = 0.9013x + 0.6935
To predict the number of fish in 12 months, we substitute x = 12 into the equation:
log y hat = 0.9013(12) + 0.6935
Simplifying the calculation:
log y hat = 10.8156 + 0.6935
log y hat = 11.5091
To find y hat (predicted value of y), we need to take the inverse logarithm (base 10) of 11.5091:
y hat = 10^11.5091
y hat ≈ 2,764,009.29
Therefore, the predicted number of fish in 12 months is approximately 2,764,009.29 fish.
To answer question 8, we need to use the linear transformation model found in question 7: 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
Simplifying the equation:
log y hat = 8.322 + 0.9013
Now, we need to solve for y hat (predicted y value). We'll take the antilog of both sides of the equation:
y hat = antilog(8.322 + 0.9013)
Using the antilog function on a calculator, we find:
y hat ≈ 52042.835
Therefore, the predicted number of fish in 12 months is approximately 52042.835.
For question 9, the power model equation is given as log y hat = 1.6 + 0.31 log x.
To determine the residual for the observed data x = 7 and y = 70, we need to calculate the predicted value (y hat) using the power model equation:
log y hat = 1.6 + 0.31 log 7
Simplifying the equation:
log y hat ≈ 1.6 + 0.31 (0.8451) ≈ 1.6 + 0.2619 ≈ 1.8619
Taking the antilog of both sides of the equation:
y hat ≈ antilog(1.8619)
Using the antilog function on a calculator, we find:
y hat ≈ 70.295
To calculate the residual, we subtract the predicted value from the observed value:
Residual = observed y - predicted y
Residual ≈ 70 - 70.295 ≈ -0.295
Therefore, the residual for the observed data x = 7 and y = 70 is approximately -0.295.