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Chapter 8: Q8E (page 325)

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. To verify the amperage of the fuses, a sample of fuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypotheses would be of interest to the manufacturer? Describe type I and type II errors in the context of this problem situation.

Short Answer

Expert verified

The hypotheses of interests are \({H_0}:\mu = 40\) versus \({H_a}:\mu \ne 40\), where \(\mu \) is the true average of amperage for the type of fuse.

Step by step solution

01

Errors in Hypothesis testing.

A type I error consists of rejecting the null hypothesis H0 when it is true.

A type II error involves not rejecting H0 when it is false.

02

Step 2:Test statistic.

A test statistic is a function of the sample data used as a basis for deciding whether H0 should be rejected. The selected test statistic should discriminate effectively between the two hypotheses. That is, values of the statistic that tend to result when H0 is true should be quite different from those typically observed when H0 is not true

03

Hypothesis results.a

The hypotheses of interests are \({H_0}:\mu = 40\) versus \({H_a}:\mu \ne 40\), where \(\mu \) is the true average of amperage for the type of fuse. Either direction is not good for the manufacturer. Therefore the alternative hypothesis should be different than \(40\).

The type I error is to conclude that \(\mu \) is not equal to \(40\) when it is \(40\).

The type II error is to conclude that \(\mu \) is \(40\) when it is different than \(40\).

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Most popular questions from this chapter

For the following pairs of assertions, indicate which with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples):

a. H0: 碌= 100, Ha: 碌 > 100

b.H0: 蟽= 20, Ha: \(\sigma \le 20\)

c.H0: p鈮� .25, Ha: p= .25

d.H0: 碌1 - 碌2 = 25, Ha: 碌1 - 碌2 > 100

e.H0: \(S_1^2 = S_2^2\) , Ha: \(S_1^2 \ne S_2^2\)

f.H0: 碌= 120, Ha: 碌= 150

g.H0: 蟽1,/蟽2 =1,Ha: 蟽1,/ 蟽2 鈮�1

h.H0p1 鈥� p2 = -.1, Ha: p1 鈥� p2 < -.1

The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in \(\overline x = 94.32\). Assume that the distribution of the melting point is normal with 蟽 =1.20.

a.Test H0: 碌 =95 versus Ha: 碌鈮� 95 using a two -tailed level .01 test.

b.If a level .01 test is used, what is 尾(94), the probability of a type II error when 碌=94?

c.What value of n is necessary to ensure that 尾(94) = .1 when 伪 = .01?

A plan for an executive travelers鈥� club has been developed by an airline on the premise that \(5\% \) of its current customers would qualify for membership. A random sample of \(500\) customers yielded \(40\) who would qualify.

a. Using this data, test at level \(.01\) the null hypothesis that the company鈥檚 premise is correct against the alternative that it is not correct.

b. What is the probability that when the test of part (a) is used, the company鈥檚 premise will be judged correct when in fact \(10\% \) of all current customers qualify?

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 \(11 - 14\) verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

\(y'' + y = tanx; y = - (cosx) ln(secx + tanx)\).
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