Information provided:
Whooping Crane Population Size 1940-1996
Year # Cranes
1940 22
1950 34
1960 33
1970 56
1980 76
1990 146
1991 132
1992 136
1993 143
1994 133
1995 158
1996 159
1.) Use exponential regression to find an exponential model of the data.
y= ae^(bx)
2.) Use your regression equation to give estimated numbers of whooping cranes in the US in 1940, 1960, and 1995. How do your results compare with the actual number?
3.) Calculate the average annual rate of increase in the whooping crane population from 1940 to 1960. Calculate the average annual rate of increase from 1960 to 1995. On which interval was the increase greater?
4.) Use your model to estimate the year in which the whooping crane population reached 100 birds.
5.) If your mode is correct, how many whooping cranes are alive in the year 2010?
-Any help is greatly appreciated!
To answer these questions, we will use exponential regression to find the exponential model of the data. Here's how we can do it step by step:
1.) Use exponential regression to find an exponential model of the data: y = ae^(bx)
- First, let's create a table using the given data:
Year | # Cranes
-------------------------
1940 | 22
1950 | 34
1960 | 33
1970 | 56
1980 | 76
1990 | 146
1991 | 132
1992 | 136
1993 | 143
1994 | 133
1995 | 158
1996 | 159
- Now, taking the natural logarithm of the equation y = ae^(bx), we get:
ln(y) = ln(a) + bx
- Let's apply this equation to our data by taking the natural logarithm of the # Cranes values. Our table becomes:
Year | ln(# Cranes)
--------------------------------
1940 | 3.091
1950 | 3.526
1960 | 3.497
1970 | 4.025
1980 | 4.331
1990 | 4.983
1991 | 4.882
1992 | 4.913
1993 | 4.962
1994 | 4.895
1995 | 5.062
1996 | 5.068
- Now, we can use a regression calculator or software to find the exponential model. By performing exponential regression on the Year and ln(# Cranes), we will obtain the values of 'a' and 'b' for the equation y = ae^(bx).
2.) Use your regression equation to give estimated numbers of whooping cranes in the US in 1940, 1960, and 1995. How do your results compare with the actual number?
- To estimate the number of whooping cranes, we will substitute the year values into the regression equation: y = ae^(bx) using the 'a' and 'b' values obtained from the regression.
- For 1940: y = ae^(bx) = a * e^(b * 1940)
- For 1960: y = ae^(bx) = a * e^(b * 1960)
- For 1995: y = ae^(bx) = a * e^(b * 1995)
- Compare these estimated values with the actual numbers given in the table to see how they compare.
3.) Calculate the average annual rate of increase in the whooping crane population from 1940 to 1960 and from 1960 to 1995. On which interval was the increase greater?
- To calculate the average annual rate of increase, we can use the formula:
Average Rate of Increase = (Ending Value / Starting Value) ^ (1 / Number of Years) - 1
- Calculate the average rate of increase for each interval using the number of cranes in the respective years and the number of years in each interval:
- For the interval 1940 to 1960,
Average Rate of Increase = (Number of Cranes in 1960 / Number of Cranes in 1940) ^ (1 / 20) - 1
- For the interval 1960 to 1995,
Average Rate of Increase = (Number of Cranes in 1995 / Number of Cranes in 1960) ^ (1 / 35) - 1
- Compare the average rates of increase to determine which interval had the greater increase.
4.) Use your model to estimate the year in which the whooping crane population reached 100 birds.
- Substitute the equation y = ae^(bx) into the model with y = 100 and solve for x.
- x represents the year in which the population reached 100 birds.
5.) If your model is correct, how many whooping cranes are alive in the year 2010?
- Substitute the equation y = ae^(bx) into the model with x = 2010 and solve for y.
- y represents the estimated number of whooping cranes alive in the year 2010.
I hope this explanation helps you solve the questions effectively!