The accompanying data on y = normalized energy \left[ (J/m^2)\right] and x = intraocular pressure (mmHg) appeared in a scatterplot in the article “Evaluating the Risk of Eye Injuries: Intraocular Pressure During High Speed Projectile Impacts” (Current Eye Research, 2012: 43–49), an estimated regression function was superimposed on the plot. x 2761 19764 25713 3980 12782 19008 y 1553 14999 32813 1667 8741 16526 x 19028 14397 9606 3905 25731 y 26770 16526 9868 6640 1220 30730 Here is Minitab output from fitting the simple linear regression model. Does the model appear to specify a useful relationship between the two variables? Predictor Coef SE Coef T P Constant -5090 2257 -2.26 0.048 Pressure 1.2912 0.1347 9.59 0.000 \left[S=3679.36, R-Sq = 90.2%, R-Sq(adj)=89.2%\right].

Question
Scatterplots
asked 2021-01-13
The accompanying data on y = normalized energy \(\displaystyle{\left[{\left(\frac{{J}}{{m}^{{2}}}\right)}\right]}\) and x = intraocular pressure (mmHg) appeared in a scatterplot in the article “Evaluating the Risk of Eye Injuries: Intraocular Pressure During High Speed Projectile Impacts” (Current Eye Research, 2012: 43–49), an estimated regression function was superimposed on the plot.
x 2761 19764 25713 3980 12782 19008 y 1553 14999 32813 1667 8741 16526 x 19028 14397 9606 3905 25731 y 26770 16526 9868 6640 1220 30730
Here is Minitab output from fitting the simple linear regression model. Does the model appear to specify a useful relationship between the two variables?
Predictor Coef SE Coef T P Constant -5090 2257 -2.26 0.048 Pressure 1.2912 0.1347 9.59 0.000
\left[S=3679.36, R-Sq = 90.2%, R-Sq(adj)=89.2%\right].

Answers (1)

2021-01-14
Step 1
Given:
\(\displaystyle{\left[{n}={12}\right]}\)
Let us assume:
\(\displaystyle{\left[\alpha={0.05}\right]}\)
Given in the output:
\(\displaystyle{\left[{b}_{{1}}={1.2912}\right]}\)
\(\displaystyle{S}{E}_{{{b}_{{{1}}}}}={0.1347}\)
Determine the hypothesis:
\(\displaystyle{H}_{{0}}:\beta_{{1}}={0}\)
\(\displaystyle{H}_{{0}}:\beta_{{1}}\ne{q}{0}\)
Compute the value of the test statistic:
\(\displaystyle{\left[{t}={\frac{{{b}_{{1}}-\beta_{{1}}}}{{{S}{E}_{{{b}_{{{1}}}}}}}}={\frac{{{1.2912}-{0}}}{{{0.1347}}}}\approx{9.59}\right]}\)
The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table B containing the t-value in the row \(\displaystyle{\left[{d}{f}={n}—{2}={12}—{2}={10}:\right]}\)
\(\displaystyle{\left[{P}{<}{2}\times{0.0005}={0.001}\right]}\)</span>
If the P-value is less than or equal to the significance level, then the null
hypothesis is rejected:
\(\displaystyle{\left[{P}{<}{0.05}\Rightarrow{R}{e}{j}{e}{c}{t}{H}_{{0}}\right]}\)</span>
There is sufficient evidence to support the claim that the slope of the population regression line is not zero, which means that the model appears to specify a useful relationship between the two variables.
Result:
Yes.
0

Relevant Questions

asked 2020-12-25
Case: Dr. Jung’s Diamonds Selection
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1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
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asked 2020-11-24
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asked 2021-02-25
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asked 2021-03-11
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\(\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.\)
\(\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(A) Complete the table below.
\(\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(Round to one decimal place as​ needed.)
\(A. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
\(B. 20602060xf(x)\)
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
\(C. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
\(D.20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
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\(\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) and for 35
\(\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\)
The mileage for the tire pressure \(\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) is
The mileage for the tire pressure \(\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}\) in. is
(Round to two decimal places as​ needed.)
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B. As tire pressure​ increases, mileage decreases.
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D. As tire pressure​ increases, mileage increases.
asked 2020-11-23
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\(\displaystyle{d}{f}={3}\)
asked 2021-02-03
The following data on = soil depth (in centimeters) and y = percentage of montmorillonite in the soil were taken from a scatterplot in the paper "Ancient Maya Drained Field Agriculture: Its Possible Application Today in the New River Floodplain, Belize, C.A." (Agricultural Ecosystems and Environment [1984]: 67-84):
a. Draw a scatterplot of y versus x.
b. The equation of the least-squares line is 0.45x. Draw this line on your scatterplot. Do there appear to be any large residuals?
c. Compute the residuals, and construct a residual plot. Are there any unusual features in the plot?
x 40 50 60 70 80 90 100
y 58 34 32 30 28 27 22
\(\displaystyle{\left[\hat{{{y}}}={64.50}\right]}\).
asked 2020-11-09
Two scatterplots are shown below.
Scatterplot 1
A scatterplot has 14 points.
The horizontal axis is labeled "x" and has values from 30 to 110.
The vertical axis is labeled "y" and has values from 30 to 110.
The points are plotted from approximately (55, 60) up and right to approximately (95, 85).
The points are somewhat scattered.
Scatterplot 2
A scatterplot has 10 points.
The horizontal axis is labeled "x" and has values from 30 to 110.
The vertical axis is labeled "y" and has values from 30 to 110.
The points are plotted from approximately (55, 55) steeply up and right to approximately (70, 90), and then steeply down and right to approximately (85, 60).
The points are somewhat scattered.
Explain why it makes sense to use the least-squares line to summarize the relationship between x and y for one of these data sets but not the other.
Scatterplot 1 seems to show a relationship between x and y, while Scatterplot 2 shows a relationship between the two variables. So it makes sense to use the least squares line to summarize the relationship between x and y for the data set in , but not for the data set in .
asked 2020-11-08
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asked 2021-01-15
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asked 2021-02-11
Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by \(\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{e}}}{N}\ {\left({8.86}\right)}.\) Solve (8.86) with the initial condition N(0) at time 0, and show that \(\displaystyle{r}_{{{e}}}\) can be estimated from \(\displaystyle{r}_{{{e}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ {\left({8.87}\right)}\)
(b) (Logistic Growth) This model is described by \(\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{l}}}{N}\ {\left({1}\ -\ {\frac{{{N}}}{{{K}}}}\right)}\ {\left({8.88}\right)}\) where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that \(\displaystyle{r}_{{{l}}}\) can be estimated from \(\displaystyle{r}_{{{l}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{K}\ -\ {N}{\left({0}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ +\ {\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{K}\ -\ {N}{\left({t}\right)}}}}\right]}\ {\left({8.89}\right)}\) for \(\displaystyle{N}{\left({t}\right)}\ {<}\ {K}.\)
(c) Assume that \(\displaystyle{N}{\left({0}\right)}={1}\) and \(\displaystyle{N}{\left({10}\right)}={1000}.\) Estimate \(\displaystyle{r}_{{{e}}}\) and \(\displaystyle{r}_{{{l}}}\) for both \(\displaystyle{K}={1001}\) and \(\displaystyle{K}={10000}.\)
(d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of \(\displaystyle{\left[{r}\right]}\) to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated.
(e) Explain why the exponential model is a good approximation to the logistic model when \(\displaystyle\frac{{N}}{{K}}\) is small compared with 1.
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