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Chapter 8: Q57SE (page 357)
In Problems \(1 - 8\) state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with \((6)\).
\(x\frac{{{d^3}y}}{{d{x^3}}} - {\left( {\frac{{dy}}{{dx}}} \right)^4} + y = 0\)
The equation is nonlinear.
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In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.
\((y - x)y' = y - x + 8;y = x + 4\sqrt {x + 2} \)
A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.
a.Define the parameter of interest and state the relevant hypotheses.
b.Suppose braking distance for the new system is normally distributed with 蟽= 10. Let \(\overline X \) denote the sample average braking distance for a random sample of 36 observations. Which values of \(\overline x \) are more contradictory to H0 than 117.2, what is the P-value in this case, and what conclusion is appropriate if 伪 = .10?
c.What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the test from part (b) is used?
A random sample of soil specimens was obtained, and the amount of organic matter (%) in the soil was determined for each specimen, resulting in the accompanying data (from 鈥淓ngineering Properties of Soil,鈥� Soil Science, 1998: 93鈥�102).
\(\begin{array}{l}1.10 5.09 0.97 1.59 4.60 0.32 0.55 1.45\\0.14 4.47 1.20 3.50 5.02 4.67 5.22 2.69\\3.98 3.17 3.03 2.21 0.69 4.47 3.31 1.17\\0.76 1.17 1.57 2.62 1.66 2.05\end{array}\)
The values of the sample mean, sample standard deviation, and (estimated) standard error of the mean are \(2.481,1.616,\) and \(.295,\) respectively. Does this data suggest that the true average percentage of organic matter in such soil is something other than \(3\% \)? Carry out a test of the appropriate hypotheses at significance level \(.10\). Would your conclusion be different if a \(\alpha = .05\) had been used? (Note: A normal probability plot of the data shows an acceptable pattern in light of the reasonably large sample size.)
Reconsider the paint-drying situation of Example 8.5, in which drying time for a test specimen is normally distributed with 蟽 = 9. The hypotheses H0: 碌 =75 versus Ha: 碌 <75 are to be tested using a random sample of n= 25 observations.
a.How many standard deviations (of X) below the null value is \(\overline x = 72.3\)?
b.If \(\overline x = 72.3\), what is the conclusion using 伪 =.002?
c.For the test procedure with 伪 =.002, what is 尾(70)?
d.If the test procedure with 伪 =.002 is used, what n is necessary to ensure that 尾(70) = .01?
e.If a level .01 test is used with n5 100, what is the probability of a type I error when m5 76?Answer the following questions for the tire problem in Example 8.7.
a.If \(\overline x = 30,960\) 30,960 and a level 伪=.01 test is used, what is the decision?
b.If a level .01 test is used, what is 尾(30,500)?
c.If a level .01 test is used and it is also required that 尾(30,500) = .05, what sample size n is necessary?
d.If \(\overline x = 30,960\), what is the smallest 伪 at which H0 can be rejected (based on n = 16)?
In Problems 21鈥�24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.
\(\frac{{{d^2}y}}{{d{x^2}}} - 4\frac{{dy}}{{dx}} + 4y = 0;y = {c_1}{e^{2x}} + {c_2}x{e^{2x}}\)
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