The authors of a paper investigated whether water temperature was related to how far a salamander would swim and whether it would swim upstream or downstream. Data for 14 streams with different mean water temperatures where salamander larvae were released are given (approximated from a graph that appeared in the paper). The two variables of interest are x = mean water temperature (°C) and y = net directionality, which was defined as the difference in the relative frequency of the released salamander larvae moving upstream and the relative frequency of released salamander larvae moving downstream. A positive value of net directionality means a higher proportion were moving upstream than downstream. A negative value of net directionality means a higher proportion were moving downstream than upstream. Mean Temperature (x) Net Directionality (y) 6.12 −0.088.01 0.25 8.57 −0.1410.61 0.00 12.5 0.08 11.94 0.03 12.55 −0.0717.93 0.29 18.24 0.23 19.84 0.24 20.3 0.19 19.12 0.14 17.78 0.05 19.57 0.07 A scatter plot of the data is given below. A scatterplot has 14 points. The horizontal axis is labeled "x" and has values from 4 to 21. The vertical axis is labeled "y" and has values from −0.2 to 0.3. 13 points are plotted fairly scattered from approximately (7, −0.14) in the lower left of the window up and right to approximately (20.5, 0.24) in the upper right of the window. 1 point is plotted at approximately (8, 0.25) in the upper left of the window. The least squares line is given below. y = −0.1412 + 0.0160xThe authors state that "when temperatures were warmer, more larvae were captured moving upstream, but when temperatures were cooler, more larvae were captured moving downstream." (a) Do the scatterplot and least-squares line support this statement? Explain. The scatterplot and least-squares line support this statement, since there is a positive relationship between mean temperature and net directionality. The scatterplot and least-squares line support this statement, since there is a negative relationship between mean temperature and net directionality. The scatterplot and least-squares line do not support this statement, since there is a negative relationship between mean temperature and net directionality. The scatterplot and least-squares line do not support this statement, since there is a positive relationship between mean temperature and net directionality. (b) Approximately what mean temperature (in °C) would result in a prediction of the same number of salamander larvae moving upstream and downstream? (Round your answer to two decimal places.) °C
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
The authors of a paper investigated whether water temperature was related to how far a salamander would swim and whether it would swim upstream or downstream. Data for 14 streams with different mean water temperatures where salamander larvae were released are given (approximated from a graph that appeared in the paper).
The two variables of interest are x = mean water temperature (°C) and y = net directionality, which was defined as the difference in the relative frequency of the released salamander larvae moving upstream and the relative frequency of released salamander larvae moving downstream. A positive value of net directionality means a higher proportion were moving upstream than downstream. A negative value of net directionality means a higher proportion were moving downstream than upstream.
Mean Temperature (x)
Net Directionality (y)
6.12
−0.08
8.01
0.25
8.57
−0.14
10.61
0.00
12.5
0.08
11.94
0.03
12.55
−0.07
17.93
0.29
18.24
0.23
19.84
0.24
20.3
0.19
19.12
0.14
17.78
0.05
19.57
0.07
A
A scatterplot has 14 points.
The horizontal axis is labeled "x" and has values from 4 to 21.
The vertical axis is labeled "y" and has values from −0.2 to 0.3.
13 points are plotted fairly scattered from approximately (7, −0.14) in the lower left of the window up and right to approximately (20.5, 0.24) in the upper right of the window.
1 point is plotted at approximately (8, 0.25) in the upper left of the window.
The least squares line is given below.
y = −0.1412 + 0.0160x
The authors state that "when temperatures were warmer, more larvae were captured moving upstream, but when temperatures were cooler, more larvae were captured moving downstream."
(a)
Do the scatterplot and least-squares line support this statement? Explain.
The scatterplot and least-squares line support this statement, since there is a positive relationship between mean temperature and net directionality.
The scatterplot and least-squares line support this statement, since there is a negative relationship between mean temperature and net directionality.
The scatterplot and least-squares line do not support this statement, since there is a negative relationship between mean temperature and net directionality.
The scatterplot and least-squares line do not support this statement, since there is a positive relationship between mean temperature and net directionality.
(b)
Approximately what mean temperature (in °C) would result in a prediction of the same number of salamander larvae moving upstream and downstream? (Round your answer to two decimal places.)
°C
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
