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Chapter 8: Q4E (page 325)

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose the specifications state that mean strength of welds should exceed 100 lb/in2 ; the inspection team decides to test H0: 碌= 100 versus Ha: 碌> 100. Explain why it might be preferable to use

this Ha rather than 碌 < 100.

Short Answer

Expert verified

The non-compatibility is better in this case because we are interested in the force required to break the weld, the non-compatible claim

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.

The statement of hypothesis \({H_0}\) is that the weld strength does not meet specifications. The hypothesis is reject when there is strong evidence that it should be rejected, evidence to the contrary.

The hypothesis \({H_a}:\mu < 100\) would yield results that the weld strength is compatible with the specifications unless proved otherwise. However, the non-compatibility is better in this case because we are interested in the force required to break the weld, the non-compatible claim.

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Most popular questions from this chapter

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 25鈥�28 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\(x\frac{{dy}}{{dx}} - 3xy = 1;y = {e^{3x}}\int_1^x {\frac{{{e^{ - 3t}}}}{t}} dt\)

Each of a group of \(20\) intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let \(p\) denote the proportion of all such players who would prefer gut to nylon, and let \(X\) be the number of players in the sample who prefer gut. Because gut strings are more expensive, consider the null hypothesis that at most \(50\% \) of all such players prefer gut. We simplify this to \({H_0}:p = .5\), planning to reject \({H_0}\) only if sample evidence strongly favors gut strings.

a. Is a significance level of exactly \(.05\) achievable? If not, what is the largest a smaller than \(.05\) that is achievable?

b. If \(60\% \) of all enthusiasts prefer gut, calculate the probability of a type II error using the significance level from part (a). Repeat if 80% of all enthusiasts prefer gut.

c. If \(13\) out of the \(20\) players prefer gut, should \({H_0}\) be rejected using the significance level of (a)?

Unlike most packaged food products, alcohol beverage container labels are not required to show calorie or nutrient content. The article 鈥淲hat Am I Drinking? The Effects of Serving Facts Information on Alcohol Beverage Containers鈥� (J. of Consumer Affairs, 2008: 81鈥�99) reported on a pilot study in which each of 58 individuals in a sample was asked to estimate the calorie content of a 12-oz can of beer known to contain 153 calories. The resulting sample mean estimated calorie level was 191 and the sample standard deviation was 89. Does this data suggest that the true average estimated calorie content in the population sampled exceeds the actual content? Test the appropriate hypotheses at significance level .001.

The true average diameter of ball bearings of a certain type is supposed to be .5 in. A one-sample t test will be carried out to see whether this is the case. What conclusion is appropriate in each of the following situations?

a.n= 13, t= 1.6, 伪= .05

b.n= 13, t= -1.6, 伪 =.05

c.n= 25, t= -2.6, 伪 =.01

d.n= 25, t= -3.9

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