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Chapter 7: Q59E (page 418)

Minimizing tractor skidding distance. Refer to the Journal of Forest Engineering (July 1999) study of minimizing tractor skidding distances along a new road in a European forest, Exercise 6.37 (p. 350). The skidding distances (in meters) were measured at 20 randomly selected road sites. The data are repeated below. Recall that a logger working on the road claims the mean skidding distance is at least 425 meters. Is there sufficient evidence to refute this claim? Use \(\alpha = .10\)

Short Answer

Expert verified

There is sufficient evidence to refute this claim.

Step by step solution

01

Given Information

The sample size is 20.

The hypothesis are given by

\(\begin{aligned}{H_0}:\mu = 425\\{H_a}:\mu > 425\end{aligned}\)

02

Compute and standard deviation

The mean is calculated as

\(\begin{aligned}\bar x &= \frac{{\sum\limits_{i = 1}^n {{X_i}} }}{n}\\ &= \frac{{7169}}{{20}}\\ &= 358.45\end{aligned}\)

The standard deviation is calculated as

\(\begin{aligned}sd &= \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({X_i} - \bar X)}^2}} }}{{n - 1}}} \\ &= \sqrt {\frac{\begin{aligned}16783.2 + 71.4025 + 9712.103 + 25424.3 + 5394.903 + 2555.303 + \\5859.903 + 46461.8 + 6488.303 + 35175 + 704.9025 + 4025.903 + \\30432.8 + 9496.503 + 7301.703 + 1726.403 + 2251.503 + 2157.603 + \\47284.5 + 4428.903\end{aligned}}{{19}}} \\ &= \sqrt {\frac{{266292.3}}{{19}}} \\ &= \sqrt {14015.38} \\ &= 118.38\end{aligned}\)

Therefore, the mean and standard deviation are 358.45 and 118.38.

03

Test statistic

The test statistic is computed as

\(\begin{aligned}t &= \frac{{\bar x - \mu }}{{\frac{s}{{\sqrt n }}}}\\ &= \frac{{358.45 - 425}}{{\frac{{118.38}}{{\sqrt {20} }}}}\\ &= \frac{{ - 66.55}}{{26.47}}\\ &= - 2.514\end{aligned}\)

Therefore, the test statistic is-2.514.

04

Conclusion

For, \(\alpha = .10\,and\,n - 1 = 19\)

\(\begin{aligned}{t_{\alpha ,n - 1}} &= {t_{0.10,19}}\\ &= 1.327\end{aligned}\)

The calculated value is less than tabulated value.

Therefore, we reject the null hypothesis.

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Most popular questions from this chapter

鈥淪treaming鈥� of television programs is trending upward. According to The Harris Poll (August 26, 2013), over one-third of American鈥檚 qualify as 鈥渟ubscription streamers,鈥� i.e., those who watch streamed TV programs through a subscription service such as Netflix, Hulu Plus, or Amazon Prime. The poll included 2,242 adult TV viewers, of which 785 are subscription streamers. On the basis of this result, can you conclude that the true fraction of adult TV viewers who are subscription streamers differs from one-third? Carry out the test using a Type I error rate of 伪=.10. Be sure to give the null and alternative hypotheses tested, test statistic value, rejection region or p-value, and conclusion.

A random sample of 175 measurements possessed a mean x炉=8.2 and a standard deviation s = .79.

a. Test H0:渭=8.3 against Ha:渭鈮�8.3Use a=0.05

Managers who engage in 鈥渃oopetition.鈥� In business, firms that both cooperate and compete with other firms are described as engaging in 鈥渃oopetition.鈥� A study published in Industrial Marketing Management (February 2016) examined the level of external tension experienced by managers who engage in coopetition. External tension (measured on a 20-point scale) was recorded for each in a sample of 1,532 managers, all from firms that were engaged in coopetition. The sample mean tension was x=10.82 and the sample standard deviation was s=3.04.

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Jury trial outcomes. Sometimes, the outcome of a jury trial defies the 鈥渃ommon sense鈥� expectations of the general public (e.g., the 1995 O. J. Simpson verdict and the 2011 Casey Anthony verdict). Such a verdict is more acceptable if we understand that the jury trial of an accused murderer is analogous to the statistical hypothesis-testing process. The null hypothesis in a jury trial is that the accused is innocent. (The status-quo hypothesis in the U.S. system of justice is innocence, which is assumed to be true until proven beyond a reasonable doubt.) The alternative hypothesis is guilt, which is accepted only when sufficient evidence exists to establish its truth. If the vote of the jury is unanimous in favor of guilt, the null hypothesis of innocence is rejected, and the court concludes that the accused murderer is guilty. Any vote other than a unanimous one for guilt results in a 鈥渘ot guilty鈥� verdict. The court never accepts the null hypothesis; that is, the court never declares the accused 鈥渋nnocent.鈥� A 鈥渘ot guilty鈥� verdict (as in the Casey Anthony case) implies that the court could not find the defendant guilty beyond a reasonable doubt

a. Define Type I and Type II errors in a murder trial.

b. Which of the two errors is the more serious? Explain.

c. The court does not, in general, know the values of 伪 and 尾 ; but ideally, both should be small. One of these probabilities is assumed to be smaller than the other in a jury trial. Which one, and why?

d. The court system relies on the belief that the value of is made very small by requiring a unanimous vote before guilt is concluded. Explain why this is so.

e. For a jury prejudiced against a guilty verdict as the trial begins, will the value of伪 increase or decrease? Explain.

f. For a jury prejudiced against a guilty verdict as the trial begins, will the value of 尾 increase or decrease? Explain

If the rejection of the null hypothesis of a particular test would cause your firm to go out of business, would you want ato be small or large? Explain
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