a scientist theorized that you can estimate the temperature by counting how often crickets chirp. the scientist gathers the data in the table shown.
A)how many cricket chirps would you expect to indicate a temperature of 85 degrees? include a graph and an equation as part of the justification of your answer
B) what might be the lowest temperature to which your model could be applied? explain your reasoning.
table shows the number of chirps in a 14-second interval: 37 32 42 37 46 35 34
temperature(degreesF): 78 72 81 77 88 75 76
do you not have a linear regression tool for finding a line of best fit? Use it, then the answers should be clearer.
Just eyeballing the data, we can first sort it:
32 72
34 76
35 75
37 77
37 78
42 81
46 88
Looks to me like I'd expect about 44 chirps at 85°
Thank you but how do you find the line of best fit?
To estimate the temperature using the number of cricket chirps, we can use the data provided in the table to analyze the relationship between the two variables.
A) Estimating temperature for 85 degrees:
We can create a scatter plot with the number of chirps on the x-axis and the temperature on the y-axis:
Number of Chirps (x) | Temperature (y)
------------------------------------
37 | 78
32 | 72
42 | 81
37 | 77
46 | 88
35 | 75
34 | 76
By observing the scatter plot, we can see that there is a positive correlation between the number of chirps and the temperature. As the temperature increases, the number of chirps generally tends to increase as well.
To estimate the temperature for 85 degrees, we can use the equation of the line of best fit. We can calculate the equation using simple linear regression:
Let's calculate the mean of the number of chirps (x) and the mean of the corresponding temperatures (y):
mean(x) = (37 + 32 + 42 + 37 + 46 + 35 + 34) / 7 = 35.57
mean(y) = (78 + 72 + 81 + 77 + 88 + 75 + 76) / 7 = 78.71
Next, let's calculate the slope (b) of the line of best fit:
b = Σ((xi - mean(x)) * (yi - mean(y))) / Σ((xi - mean(x))^2)
Using the data from the table, we can calculate the slope as follows:
Σ((xi - mean(x)) * (yi - mean(y))) = (37 - 35.57) * (78 - 78.71) + (32 - 35.57) * (72 - 78.71) + (42 - 35.57) * (81 - 78.71) + (37 - 35.57) * (77 - 78.71) + (46 - 35.57) * (88 - 78.71) + (35 - 35.57) * (75 - 78.71) + (34 - 35.57) * (76 - 78.71)
= -31.70
Σ((xi - mean(x))^2) = (37 - 35.57)^2 + (32 - 35.57)^2 + (42 - 35.57)^2 + (37 - 35.57)^2 + (46 - 35.57)^2 + (35 - 35.57)^2 + (34 - 35.57)^2
= 131.43
b = -31.70 / 131.43
≈ -0.2412
Now, let's calculate the y-intercept (a) of the line of best fit:
a = mean(y) - b * mean(x)
= 78.71 - (-0.2412 * 35.57)
≈ 87.20
Therefore, the equation of the line of best fit is:
y = -0.2412x + 87.20
To estimate the temperature for 85 degrees, we can substitute x = 85 into the equation:
y = -0.2412(85) + 87.20
≈ 66.78
Based on the model, we would expect around 66.78 chirps for a temperature of approximately 85 degrees.
B) The lowest temperature to which this model could be applied:
Looking at the given data, we can see that the lowest temperature recorded is 72 degrees (corresponding to 32 chirps). Below 72 degrees, we do not have any data points, so it is difficult to accurately estimate the temperature using this model. Therefore, it would be reasonable to assume that the model may not be reliable for temperatures below 72 degrees.
To estimate the temperature using the number of chirps, we need to find a relationship between the temperature and the chirp count. To do this, we can start by creating a scatter plot of the data points given in the table.
First, we will plot the number of chirps on the x-axis and the corresponding temperature on the y-axis. The data points to be used for this are:
Chirps: 37, 32, 42, 37, 46, 35, 34
Temperature: 78, 72, 81, 77, 88, 75, 76
A scatter plot will help visualize any pattern or relationship between the two variables.
Now, we draw a scatter plot:
(37, 78)
(32, 72)
(42, 81)
(37, 77)
(46, 88)
(35, 75)
(34, 76)
Next, we'll sketch the scatter plot graph.
Temperature
|
88 *
|
86 | *
|
84 |
|
82 |
|
80 |
|
78 | *
|
76 |
|
74 |
+----+----+----+
32 37 42 46 Chirps
From the scatter plot, we can see that there appears to be a positive relationship between the temperature and the number of cricket chirps. As the number of chirps increases, the temperature generally tends to increase as well.
To estimate the temperature for a given number of chirps, we can find an equation for the line of best fit. A simple linear equation can be used to model this relationship:
Temperature = m * Chirps + c
To find the equation of the line of best fit, we can use regression analysis or perform a linear regression. However, since we have a small amount of data, we can estimate the equation roughly by drawing a straight line on the scatter plot that seems to fit the general direction of the data points.
Let's draw a line that roughly fits the points:
Temperature
|
88 *
/
86 /
/
84 /
/
82 /
/
80 --
x
78 x
x
76 x
+----+----+----+
32 37 42 46 Chirps
The equation for this line can be approximated as:
Temperature ≈ 2 * Chirps + 40
This equation is just an approximation and may not be the exact equation for the relationship. However, it provides us with a rough estimate of the temperature based on the number of chirps.
A) To estimate the temperature for 85 degrees, we can plug in the value of "85" for Temperature in the equation:
85 ≈ 2 * Chirps + 40
Now, we can solve for Chirps:
85 - 40 ≈ 2 * Chirps
45 ≈ 2 * Chirps
Chirps ≈ 22.5
We can round the chirp count to the nearest whole number, so we would expect around 23 chirps to indicate a temperature of 85 degrees.
B) Regarding the lowest temperature to which the model can be applied, we need to consider the range of the given data points. From the table, the lowest temperature is 72 degrees (corresponding to 32 chirps).
While the model may estimate temperatures lower than 72 degrees, it would be less reliable as it is extrapolating beyond the range of observed data. Therefore, the lowest temperature to which the model can be confidently applied is 72 degrees.