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Chapter 14: Problem 16

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 exact P-values are dependent on the specifics of the F-table or statistics software employed. Nonetheless, for each scenario, the P-value represents the probability that the F-statistic is larger than the calculated value given the degrees of freedom. Always remember, a lower P-value suggests stronger statistical evidence against the null hypothesis.

Step by step solution

01

Understanding the F-statistic

For a given F-statistic and its degrees of freedom, we can obtain the respective P-value. The calculated F-statistic is derived from the variances of two independent samples. The degrees of freedom are given as \(\mathrm{df}_{1}\) and \(\mathrm{df}_{2}\).
02

Use an F-table or a statistics software to find P-value for \(\mathrm{df}_{1}=3, \mathrm{df}_{2}=15\), calculated \(F=4.23\)

The desired P-value is the probability that the F-statistic is larger than the calculated value of 4.23. This probability can be obtained from an F-table or a statistics software by inputting the degrees of freedom and the F-statistic. The exact value may not be found on an F-table, but a range can be estimated.
03

Repeat step 2 for each scenario

Repeat the same process to get the P-value for the other scenarios in b, c, d with different degrees of freedom and F-statistics. Recall that a lower P-value suggests a lower probability of the observed differences happening due to chance, hence stronger evidence against the null hypothesis.

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Most popular questions from this chapter

This exercise requires the use of a computer package. The accompanying data resulted from a study of the relationship between \(y=\) brightness of finished paper and the independent variables \(x_{1}=\) hydrogen peroxide (\% by weight), \(x_{2}=\) sodium hydroxide (\% by weight), \(x_{3}=\) silicate \((\%\) by weight \()\), and \(x_{4}=\) process temperature ("Advantages of CE-HDP Bleaching for High Brightness Kraft Pulp Production," TAPPI [1964]: 107A-173A). $$ \begin{array}{ccccc} x_{1} & x_{2} & x_{3} & x_{4} & y \\ \hline .2 & .2 & 1.5 & 145 & 83.9 \\ .4 & .2 & 1.5 & 145 & 84.9 \\ .2 & .4 & 1.5 & 145 & 83.4 \\ .4 & .4 & 1.5 & 145 & 84.2 \\ .2 & .2 & 3.5 & 145 & 83.8 \\ .4 & .2 & 3.5 & 145 & 84.7 \\ .2 & .4 & 3.5 & 145 & 84.0 \\ .4 & .4 & 3.5 & 145 & 84.8 \\ .2 & .2 & 1.5 & 175 & 84.5 \\ .4 & .2 & 1.5 & 175 & 86.0 \\ .2 & .4 & 1.5 & 175 & 82.6 \\ .4 & .4 & 1.5 & 175 & 85.1 \\ .2 & .2 & 3.5 & 175 & 84.5 \\ .4 & .2 & 3.5 & 175 & 86.0 \\ .2 & .4 & 3.5 & 175 & 84.0 \\ .4 & .4 & 3.5 & 175 & 85.4 \\ .1 & .3 & 2.5 & 160 & 82.9 \\ .5 & .3 & 2.5 & 160 & 85.5\\\ .3 & .1 & 2.5 & 160 & 85.2 \\ .3 & .5 & 2.5 & 160 & 84.5 \\ .3 & .3 & 0.5 & 160 & 84.7 \\ .3 & .3 & 4.5 & 160 & 85.0 \\ .3 & .3 & 2.5 & 130 & 84.9 \\ .3 & .3 & 2.5 & 190 & 84.0 \\ .3 & .3 & 2.5 & 160 & 84.5 \\ .3 & .3 & 2.5 & 160 & 84.7 \\ .3 & .3 & 2.5 & 160 & 84.6 \\ .3 & .3 & 2.5 & 160 & 84.9 \\ .3 & .3 & 2.5 & 160 & 84.9 \\ .3 & .3 & 2.5 & 160 & 84.5 \\ .3 & .3 & 2.5 & 160 & 84.6 \end{array} $$ a. Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. b. Using a \(.05\) significance level, perform the model utility test. c. Interpret the values of the following quantities: SSResid, \(R^{2}, s_{e}\)

The article "The Undrained Strength of Some Thawed Permafrost Soils" (Canadian Geotechnical Journal \([1979]: 420-427\) ) contained the accompanying data (see page 778 ) on \(y=\) shear strength of sandy soil \((\mathrm{kPa})\), \(x_{1}=\) depth \((\mathrm{m})\), and \(x_{2}=\) water content \((\%) .\) The predicted values and residuals were computed using the estimated regression equation $$ \begin{aligned} \hat{y}=&-151.36-16.22 x_{1}+13.48 x_{2}+.094 x_{3}-.253 x_{4} \\ &+.492 x_{5} \\ \text { where } x_{3} &=x_{1}^{2}, x_{4}=x_{2}^{2}, \text { and } x_{5}=x_{1} x_{2} \end{aligned} $$ $$ \begin{array}{clrrrrr} \text { Product } & \text { Material } & \text { Height } & \begin{array}{l} \text { Maximum } \\ \text { Width } \end{array} & \begin{array}{l} \text { Minimum } \\ \text { Width } \end{array} & \text { Elongation } & \text { Volume } \\ \hline 1 & \text { glass } & 7.7 & 2.50 & 1.80 & 1.50 & 125 \\ 2 & \text { glass } & 6.2 & 2.90 & 2.70 & 1.07 & 135 \\ 3 & \text { glass } & 8.5 & 2.15 & 2.00 & 1.98 & 175 \\ 4 & \text { glass } & 10.4 & 2.90 & 2.60 & 1.79 & 285 \\ 5 & \text { plastic } & 8.0 & 3.20 & 3.15 & 1.25 & 330 \\ 6 & \text { glass } & 8.7 & 2.00 & 1.80 & 2.17 & 90 \\ 7 & \text { glass } & 10.2 & 1.60 & 1.50 & 3.19 & 120 \\ 8 & \text { plastic } & 10.5 & 4.80 & 3.80 & 1.09 & 520 \\ 9 & \text { plastic } & 3.4 & 5.90 & 5.00 & 0.29 & 330 \\ 10 & \text { plastic } & 6.9 & 5.80 & 4.75 & 0.59 & 570\\\ 11 & \text { tin } & 10.9 & 2.90 & 2.80 & 1.88 & 340 \\ 12 & \text { plastic } & 9.7 & 2.45 & 2.10 & 1.98 & 175 \\ 13 & \text { glass } & 10.1 & 2.60 & 2.20 & 1.94 & 240 \\ 14 & \text { glass } & 13.0 & 2.60 & 2.60 & 2.50 & 240 \\ 15 & \text { glass } & 13.0 & 2.70 & 2.60 & 2.41 & 360 \\ 16 & \text { glass } & 11.0 & 3.10 & 2.90 & 1.77 & 310 \\ 17 & \text { cardboard } & 8.7 & 5.10 & 5.10 & 0.85 & 635 \\ 18 & \text { cardboard } & 17.1 & 10.20 & 10.20 & 0.84 & 1250 \\ 19 & \text { glass } & 16.5 & 3.50 & 3.50 & 2.36 & 650 \\ 20 & \text { glass } & 16.5 & 2.70 & 1.20 & 3.06 & 305 \\ 21 & \text { glass } & 9.7 & 3.00 & 1.70 & 1.62 & 315 \\ 22 & \text { glass } & 17.8 & 2.70 & 1.75 & 3.30 & 305 \\ 23 & \text { glass } & 14.0 & 2.50 & 1.70 & 2.80 & 245 \\ 24 & \text { glass } & 13.6 & 2.40 & 1.20 & 2.83 & 200 \\ 25 & \text { plastic } & 27.9 & 4.40 & 1.20 & 3.17 & 1205 \\ 26 & \text { tin } & 19.5 & 7.50 & 7.50 & 1.30 & 2330 \\ 27 & \text { tin } & 13.8 & 4.25 & 4.25 & 1.62 & 730 \end{array} $$ $$ \begin{array}{rrrrr} {\boldsymbol{y}} & {\boldsymbol{x}_{1}} & \boldsymbol{x}_{2} & \text { Predicted } \boldsymbol{y} & {\text { Residual }} \\ \hline 14.7 & 8.9 & 31.5 & 23.35 & -8.65 \\ 48.0 & 36.6 & 27.0 & 46.38 & 1.62 \\ 25.6 & 36.8 & 25.9 & 27.13 & -1.53 \\ 10.0 & 6.1 & 39.1 & 10.99 & -0.99 \\ 16.0 & 6.9 & 39.2 & 14.10 & 1.90 \\ 16.8 & 6.9 & 38.3 & 16.54 & 0.26 \\ 20.7 & 7.3 & 33.9 & 23.34 & -2.64 \\ 38.8 & 8.4 & 33.8 & 25.43 & 13.37 \\ 16.9 & 6.5 & 27.9 & 15.63 & 1.27 \\ 27.0 & 8.0 & 33.1 & 24.29 & 2.71 \\ 16.0 & 4.5 & 26.3 & 15.36 & 0.64 \\ 24.9 & 9.9 & 37.8 & 29.61 & -4.71 \\ 7.3 & 2.9 & 34.6 & 15.38 & -8.08 \\ 12.8 & 2.0 & 36.4 & 7.96 & 4.84 \\ \hline \end{array} $$ a. Use the given information to compute SSResid, SSTo, and SSRegr. b. Calculate \(R^{2}\) for this regression model. How would you interpret this value? c. Use the value of \(R^{2}\) from Part (b) and a .05 level of significance to conduct the appropriate model utility test.

The article "Impacts of On-Campus and Off-Campus Work on First-Year Cognitive Outcomes" (Journal of College Student Development \([1994]: 364-370\) ) reported on a study in which \(y=\) spring math comprehension score was regressed against \(x_{1}=\) previous fall test score, \(x_{2}=\) previous fall academic motivation, \(x_{3}=\) age, \(x_{4}=\) number of credit hours, \(x_{5}=\) residence \((1\) if on campus, 0 otherwise), \(x_{6}=\) hours worked on campus, and \(x_{7}=\) hours worked off campus. The sample size was \(n=210\), and \(R^{2}=.543\). Test to see whether there is a useful linear relationship between \(y\) and at least one of the predictors.

This exercise requires the use of a computer package. The cotton aphid poses a threat to cotton crops in Iraq. The accompanying data on \(y=\) infestation rate (aphids/100 leaves) \(x_{1}=\) mean temperature \(\left({ }^{\circ} \mathrm{C}\right)\) \(x_{2}=\) mean relative humidity appeared in the article "Estimation of the Economic Threshold of Infestation for Cotton Aphid" (Mesopotamia Journal of Agriculture [1982]: 71-75). Use the data to find the estimated regression equation and assess the utility of the multiple regression model $$ y=\alpha+\beta_{1} x_{1}+\beta_{2} x_{2}+e $$ $$ \begin{array}{rrrrrr} \boldsymbol{y} & \boldsymbol{x}_{1} & \boldsymbol{x}_{2} & \boldsymbol{y} & \boldsymbol{x}_{1} & \boldsymbol{x}_{2} \\ \hline 61 & 21.0 & 57.0 & 77 & 24.8 & 48.0 \\ 87 & 28.3 & 41.5 & 93 & 26.0 & 56.0 \\ 98 & 27.5 & 58.0 & 100 & 27.1 & 31.0 \\ 104 & 26.8 & 36.5 & 118 & 29.0 & 41.0 \\ 102 & 28.3 & 40.0 & 74 & 34.0 & 25.0 \\ 63 & 30.5 & 34.0 & 43 & 28.3 & 13.0 \\ 27 & 30.8 & 37.0 & 19 & 31.0 & 19.0\\\ 14 & 33.6 & 20.0 & 23 & 31.8 & 17.0 \\ 30 & 31.3 & 21.0 & 25 & 33.5 & 18.5 \\ 67 & 33.0 & 24.5 & 40 & 34.5 & 16.0 \\ 6 & 34.3 & 6.0 & 21 & 34.3 & 26.0 \\ 18 & 33.0 & 21.0 & 23 & 26.5 & 26.0 \\ 42 & 32.0 & 28.0 & 56 & 27.3 & 24.5 \\ 60 & 27.8 & 39.0 & 59 & 25.8 & 29.0 \\ 82 & 25.0 & 41.0 & 89 & 18.5 & 53.5 \\ 77 & 26.0 & 51.0 & 102 & 19.0 & 48.0 \\ 108 & 18.0 & 70.0 & 97 & 16.3 & 79.5 \end{array} $$

Suppose that the variables \(y, x_{1}\), and \(x_{2}\) are related by the regression model $$ y=1.8+.1 x_{1}+.8 x_{2}+e $$ a. Construct a graph (similar to that of Figure \(14.5)\) showing the relationship between mean \(y\) and \(x_{2}\) for fixed values 10,20 , and 30 of \(x_{1}\). b. Construct a graph depicting the relationship between mean \(y\) and \(x_{1}\) for fixed values 50,55, and 60 of \(x_{2}\). c. What aspect of the graphs in Parts (a) and (b) can be attributed to the lack of an interaction between \(x_{1}\) and \(x_{2}\) ? d. Suppose the interaction term \(.03 x_{3}\) where \(x_{3}=x_{1} x_{2}\) is added to the regression model equation. Using this new model, construct the graphs described in Parts (a) and (b). How do they differ from those obtained in Parts (a) and (b)?
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