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 −0.088.11 0.25 8.57 −0.1410.51 0.00 12.5 0.08 12.04 0.03 12.45 −0.0717.93 0.29 18.34 0.23 19.84 0.24 20.3 0.19 19.02 0.14 17.78 0.05 19.67 0.07 (a) Construct a scatterplot of the data. 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 (4.5, 0.24) in the upper left of the window down and right to approximately (18, −0.14) in the lower right of the window.1 point is plotted at approximately (17, 0.25) in the upper right of the window.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, 0.24) in the upper left of the window down and right to approximately (20.5, −0.14) in the lower right of the window.1 point is plotted at approximately (8, −0.15) in the lower left of the window.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 (4.5, −0.14) in the lower left of the window up and right to approximately (18, 0.24) in the upper right of the window.1 point is plotted at approximately (17, −0.15) in the lower right of the window.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. How would you describe the relationship between x and y? There is an outlier in the upper right of the window. If we disregard that outlier, then there is a negative linear relationship between mean temperature and net directionality. There is an outlier in the lower right of the window. If we disregard that outlier, then there is a positive linear relationship between mean temperature and net directionality. There is an outlier in the lower left of the window. If we disregard that outlier, then there is a negative linear relationship between mean temperature and net directionality. There is an outlier in the upper left of the window. If we disregard that outlier, then there is a positive linear relationship between mean temperature and net directionality. (b) Find the equation of the least-squares line describing the relationship between y = net directionality and x = mean water temperature. (Round your values to five decimal places.) ŷ = + x(c) What value of net directionality would you predict for a stream that had mean water temperature of 14°C? (Round your answer to five decimal places.)
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
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
8.11
0.25
8.57
−0.14
10.51
0.00
12.5
0.08
12.04
0.03
12.45
−0.07
17.93
0.29
18.34
0.23
19.84
0.24
20.3
0.19
19.02
0.14
17.78
0.05
19.67
0.07
(a)
Construct 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 (4.5, 0.24) in the upper left of the window down and right to approximately (18, −0.14) in the lower right of the window.
1 point is plotted at approximately (17, 0.25) in the upper right of the window.
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, 0.24) in the upper left of the window down and right to approximately (20.5, −0.14) in the lower right of the window.
1 point is plotted at approximately (8, −0.15) in the lower left of the window.
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 (4.5, −0.14) in the lower left of the window up and right to approximately (18, 0.24) in the upper right of the window.
1 point is plotted at approximately (17, −0.15) in the lower right of the window.
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.
How would you describe the relationship between x and y?
There is an outlier in the upper right of the window. If we disregard that outlier, then there is a negative linear relationship between mean temperature and net directionality.
There is an outlier in the lower right of the window. If we disregard that outlier, then there is a positive linear relationship between mean temperature and net directionality.
There is an outlier in the lower left of the window. If we disregard that outlier, then there is a negative linear relationship between mean temperature and net directionality.
There is an outlier in the upper left of the window. If we disregard that outlier, then there is a positive linear relationship between mean temperature and net directionality.
(b)
Find the equation of the least-squares line describing the relationship between y = net directionality and x = mean water temperature. (Round your values to five decimal places.)
ŷ = +
x
(c)
What value of net directionality would you predict for a stream that had mean water temperature of 14°C? (Round your answer to five decimal places.)
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