11.4: Biometrics Lab #4
- Page ID
- 2942
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Name: ______________________________________________________
Experiment 1
The following data were collected on Old Faithful geyser in Yellowstone Park. The x-variable is time between eruptions and the y-variable is length of eruptions.
|
X |
Y |
|---|---|
|
12.17 |
1.88 |
|
11.63 |
1.77 |
|
12.03 |
1.83 |
|
12.15 |
1.83 |
|
11.30 |
1.70 |
|
11.70 |
1.82 |
|
12.27 |
1.93 |
|
11.60 |
1.77 |
|
11.72 |
1.83 |
|
12.10 |
1.89 |
|
11.70 |
1.80 |
|
11.40 |
1.72 |
|
11.22 |
1.75 |
|
11.42 |
1.73 |
|
11.53 |
1.74 |
|
11.50 |
1.77 |
|
11.90 |
1.87 |
|
11.86 |
1.84 |
a) Determine if a relationship exists between the 2 variables using a scatterplot and the linear correlation coefficient. Select Graph> Scatterplot. Select the Simple plot and click OK. Enter the response variable (length of eruptions) in the Y variables box, and the predictor variable (time between eruptions) in the X variables box. Click OK. Describe the relationship that you see.
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b) Calculate the linear correlation coefficient. Statistics> Basic Stats> Correlation. Enter the 2 variables in the Variables box and click OK.
r = ____________________________________
What two pieces of information about the relationship between these two variables does the linear correlation coefficient tell you?
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c) Find a least squares regression line treating “time between eruptions” as the predictor variable (x) and “length of eruptions” as the response variable (y). Stat>Regression> General Regression.Enter “length of eruptions” in the Response box. Enter “time between eruptions” in the Model box. Click on Options and make sure that 95% is selected for all confidence intervals. Click on Graphsand select the Residual plot “Residual versus fits.” Click Results and make sure the Regression equation, Coefficient table, Display confidence intervals, Summary of model, Analysis of Variance table, and prediction tables are checked. Click OK.
Write the regression equation __________________
What is the value of R2? _______________________
What does this mean?
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Examine the residual model. Do you see any problems?
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What is the value of the regression standard error? _____________________________
Write the confidence intervals for the y-intercept ______________________________
and slope ______________________________________________________________
Use the output to test if the slope is significantly different from zero. Write the null and alternative hypotheses for this test.
H0:____________________________________
H1: ____________________________________
Using the test statistic and p-value from the Minitab output to test this claim.
Test statistic_______________________________ p-value _______________________________
Conclusion:
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d) Using the regression equation, what would be the length of the eruption if the time between eruptions is 11.42 min.?
Experiment 2
The index of biotic integrity (IBI) is a measure of water quality in streams. The sample data given in the table below comes from the Piedmont forest region. The table gives the data for IBI and forested area in square kilometers. Let Forest Area be the predictor variable (x) and IBI be the response variable (y).
Create a scatterplot and describe the relationship between these variables. Compute the linear correlation coefficient.
r = ____________________________________
Create a regression model for this data set following the steps from the first example. Write the regression model.
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Is there significant evidence to support the claim that IBI increases with Forest Area? Write the test statistic/p-value used for this slope test along with your answer.
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The researcher wants to estimate the population mean IBI for streams that have an average forested area of 48 sq. km. Click STAT>REGRESSION> GENERAL REGRESSION. Making sure that IBI is in the Response box and Forest Area is in the Model box, click on Prediction and enter 48 in the New observation for continuous predictors box and check Confidence limits. Click OK. Write the 95% confidence interval for mean IBI for streams in an average forested area of 48 sq. km. ______________________________________________________
You are working with a stream in an area with 19 sq. km. of forested area. Your management plan includes an afforestation project that will increase the forested area to 23 sq. km. You need to predict what the specific IBI would be for this stream when the forested area is increased. Create a prediction interval to estimate this IBI if the forested area increased to 23 sq. km.
Click STAT>REGRESSION>GENERAL REGRESSION. Making sure that IBI is in the Response box and Forest Area is in the Model box, click on Prediction and enter 23 in the New observation for continuous predictors box and check Prediction limits. Click OK. Write the 95% prediction interval for the IBI for this stream when the forested area is increased to 23 sq. km. ___________________________________________________
Explain the difference between the confidence and prediction intervals you just computed.
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