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Chapter 14: Problem 17

Obtain as much information as you can about the \(P\) -value for an upper-tailed \(F\) test in each of the following situations: a. \(\mathrm{df}_{1}=3, \mathrm{df}_{2}=15\), calculated \(F=4.23\) b. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=18\), calculated \(F=1.95\) c. \(\mathrm{df}_{1}=5, \mathrm{df}_{2}=20\), calculated \(F=4.10\) d. \(\mathrm{df}_{1}=4, \mathrm{df}_{2}=35\), calculated \(F=4.58\)

Short Answer

Expert verified
The P-value for each situation can be found using an F-distribution table or calculator with the given degree of freedom of the numerator, degree of freedom of the denominator, and the calculated F value. The exact P-values depend on the specific table or calculator used and so are not provided here.

Step by step solution

01

Find the P-value for the first situation

First, identify the given values. Thus, \(\mathrm{df}_{1}=3\), \(\mathrm{df}_{2}=15\), and calculated \(F=4.23\). Then, using an F-distribution table or calculator, find the P-value corresponding to these values.
02

Find the P-value for the second situation

For this situation, we have \(\mathrm{df}_{1}=4\), \(\mathrm{df}_{2}=18\), and calculated \(F=1.95\). Following a similar process as in step 1, we use an F-distribution table or calculator to find the corresponding P-value.
03

Find the P-value for the third situation

Given are \(\mathrm{df}_{1}=5\), \(\mathrm{df}_{2}=20\), and calculated \(F=4.10\). We apply the same process as the previous steps using the F-distribution table or calculator to find the associated P-value.
04

Find the P-value for the fourth situation

In the fourth situation, the values are \(\mathrm{df}_{1}=4\), \(\mathrm{df}_{2}=35\), and calculated \(F=4.58\). Using the table or calculator for the F-distribution, we find the corresponding P-value.

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Most popular questions from this chapter

The accompanying Minitab output results from fitting the model described in Exercise \(14.14\) to data. $$ \begin{aligned} &\begin{array}{lrrr} \text { Predictor } & \text { Coet } & \text { Stdev } & \text { t-ratio } \\ \text { Constant } & 86.85 & 85.39 & 1.02 \\ \text { X1 } & -0.12297 & 0.03276 & -3.75 \\ \text { X2 } & 5.090 & 1.969 & 2.58 \\ \text { X3 } & -0.07092 & 0.01799 & -3.94 \\ \text { X }_{4} & 0.0015380 & 0.0005560 & 2.77 \\ \mathrm{~S}=4.784 & \text { R-sq }=90.8 \% & R-\mathrm{sq}(\mathrm{adj})=89.4 \% \end{array}\\\ &\text { Analysis of Variance }\\\ &\begin{array}{lrrr} & \text { DF } & \text { SS } & \text { MS } \\ \text { Regression } & 4 & 5896.6 & 1474.2 \\ \text { Error } & 26 & 595.1 & 22.9 \\ \text { Total } & 30 & 6491.7 & \end{array} \end{aligned} $$ a. What is the estimated regression equation? b. Using a \(.01\) significance level, perform the model utility test. c. Interpret the values of \(R^{2}\) and \(s_{e}\) given in the output.

When coastal power stations take in large quantities of cooling water, it is inevitable that a number of fish are drawn in with the water. Various methods have been designed to screen out the fish. The article "Multiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes" (Journal of Applied Ecology [1983]: 33-42) examined intake fish catch at an English power plant and several other variables thought to affect fish intake: $$ \begin{aligned} y &=\text { fish intake (number of fish) } \\ x_{1} &=\text { water temperature }\left({ }^{\circ} \mathrm{C}\right) \\ x_{2} &=\text { number of pumps running } \\ x_{3} &=\text { sea state }(\text { values } 0,1,2, \text { or } 3) \\ x_{4} &=\text { speed (knots) } \end{aligned} $$ Part of the data given in the article were used to obtain the estimated regression equation $$ \hat{y}=92-2.18 x_{1}-19.20 x_{2}-9.38 x_{3}+2.32 x_{4} $$ (based on \(n=26\) ). SSRegr \(=1486.9\) and SSResid \(=\) \(2230.2\) were also calculated. a. Interpret the values of \(b_{1}\) and \(b_{4}\). b. What proportion of observed variation in fish intake can be explained by the model relationship? c. Estimate the value of \(\sigma\). d. Calculate adjusted \(R^{2}\). How does it compare to \(R^{2}\) itself?

The ability of ecologists to identify regions of greatest species richness could have an impact on the preservation of genetic diversity, a major objective of the World Conservation Strategy. The article "Prediction of Rarities from Habitat Variables: Coastal Plain Plants on Nova Scotian Lakeshores" (Ecology [1992]: 1852 - 1859) used a sample of \(n=37\) lakes to obtain the estimated regression equation \(\begin{aligned} \hat{y}=& 3.89+.033 x_{1}+.024 x_{2}+.023 x_{3} \\ &+.008 x_{4}-.13 x_{5}-.72 x_{6} \end{aligned}\) where \(y=\) species richness, \(x_{1}=\) watershed area, \(x_{2}=\) shore width, \(x_{3}=\) drainage \((\%), x_{i}=\) water color (total color units), \(x_{5}=\) sand \((\%)\), and \(x_{6}=\) alkalinity. The coefficient of multiple determination was reported as \(R^{2}=.83 .\) Use a test with significance level \(.01\) to decide whether the chosen model is useful.

This exercise requires the use of a computer package. The paper "Habitat Selection by Black Bears in an Intensively Logged Boreal Forrest" (Canadian Journal of Zoology [2008]: \(1307-1316\) ) gave the accompanying data on \(n=11\) female black bears. $$ \begin{array}{ccc} \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Weight } \\ (\mathrm{kg}) \end{array} & \begin{array}{c} \text { Home-Range } \\ \text { Size }\left(\mathrm{km}^{2}\right) \end{array} \\ \hline 10.5 & 54 & 43.1 \\ 6.5 & 40 & 46.6 \\ 28.5 & 62 & 57.4 \\ 6.5 & 55 & 35.6 \\ 7.5 & 56 & 62.1 \\ 6.5 & 62 & 33.9 \\ 5.5 & 42 & 39.6 \\ 7.5 & 40 & 32.2 \\ 11.5 & 59 & 57.2 \\ 9.5 & 51 & 24.4 \\ 5.5 & 50 & 68.7 \\ \hline \end{array} $$ a. Fit a multiple regression model to describe the relationship between \(y=\) home-range size and the predictors \(x_{1}=\) age and \(x_{2}=\) weight. b. Construct a normal probability plot of the 11 standardized residuals. Based on the plot, does it seem reasonable to regard the random deviation distribution as approximately normal? Explain. c. If appropriate, carry out a model utility test with a significance level of \(.05\) to determine if the predictors age and weight are useful for predicting homerange size.

The article "The Value and the Limitations of High-Speed Turbo-Exhausters for the Removal of Tar-Fog from Carburetted Water-Gas" (Society of Chemical Industry Journal [1946]: \(166-168\) ) presented data on \(y=\) tar content \(\left(\right.\) grains \(\left./ 100 \mathrm{ft}^{3}\right)\) of a gas stream as a function of \(x_{1}=\) rotor speed (rev/minute) and \(x_{2}=\) gas inlet temperature \(\left({ }^{\circ} \mathrm{F}\right)\). A regression model using \(x_{1}, x_{2}, x_{3}=x_{2}^{2}\) and \(x_{i}=x_{1} x_{2}\) was suggested: $$ \begin{aligned} \text { mean } y \text { value }=& 86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} \\ &+.001 x_{i} \end{aligned} $$ a. According to this model, what is the mean \(y\) value if \(x_{1}=3200\) and \(x_{2}=57 ?\) b. For this particular model, does it make sense to interpret the value of a \(\beta_{2}\) as the average change in tar content associated with a 1 -degree increase in gas inlet temperature when rotor speed is held constant? Explain.
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