/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q6E Let 碌 denote the true average r... [FREE SOLUTION] | 魅影直播

魅影直播

Log out

Learning Materials

Features

Discover

Chapter 8: Q6E (page 325)

Let 碌 denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing H0: 碌= 5 versus Ha: 碌> 5 or H0: 碌= 5 versus Ha: 碌 < 5? Explain your reasoning. (Hint: Think about the consequences of a type I and type II error for each possibility.)

Short Answer

Expert verified

Yes , I recommend\({H_0}:\mu = 5\)versus\({H_a}:\mu > 5\).

It is better to choose that the type I error is more serious so that we can control the error(explicitly).

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.

Usually, you would like for a type I error to be very small (highly unlikely). In this case, when the alternative hypothesis is \({H_a}:\mu < 5\), the type I error is deciding that water is safe when it is not- this can affect a lot of people and it is a grave error. The type II errors says that the water is unsafe when it is safe, this error is not as serious as the first there are no such consequences.

In the second case, when \({H_a}:\mu > 5\), the type II error is now that the water is safe while it is not, which is the more serious case. As mentioned in the beginning, it is better to choose that the type I error is more serious so that we can control the error(explicitly).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 魅影直播!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The following observations are on stopping distance (ft) of a particular truck at \(20mph\) under specified experimental conditions (鈥淓xperimental Measurement of the Stopping Performance of a Tractor-Semitrailer from Multiple Speeds,鈥� NHTSA, DOT HS 811 488, June 2011):

\(32.1 30.6 31.4 30.4 31.0 31.9\)

The cited report states that under these conditions, the maximum allowable stopping distance is \(30\). A normal probability plot validates the assumption that stopping distance is normally distributed.

a. Does the data suggest that true average stopping distance exceeds this maximum value? Test the appropriate hypotheses using \(\alpha = .01\).

b. Determine the probability of a type II error when a 5 .01, \(\sigma = .65\), and the actual value of \(\mu \) is \(31\). Repeat this for \(\mu = 32\) (use either statistical software or Table A.17).

c. Repeat (b) using \(\sigma = .80\) and compare to the results of (b).

d. What sample size would be necessary to have \(\alpha = .01\) and \(\beta = .10\) when \(\mu = 31\) and \(\sigma = .65\)?

The accompanying data on cube compressive strength (MPa) of concrete specimens appeared in the article 鈥淓xperimental Study of Recycled Rubber-Filled High-Strength Concrete鈥� (Magazine of Concrete Res., 2009: 549鈥�556):

\(\begin{array}{l}112.3 97.0 92.7 86.0 102.0\\99.2 95.8 103.5 89.0 86.7\end{array}\)

a. Is it plausible that the compressive strength for this type of concrete is normally distributed?

b. Suppose the concrete will be used for a particular application unless there is strong evidence that true average strength is less than \(100MPa\). Should the concrete be used? Carry out a test of appropriate hypotheses.

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

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)?

Have you ever been frustrated because you could not get a container of some sort to release the last bit of its contents? The article 鈥淪hake, Rattle, and Squeeze: How Much Is Left in That Container?鈥� (Consumer Reports, May 2009: 8) reported on an investigation of this issue for various consumer products. Suppose five \(6.0oz\) tubes of toothpaste of a particular brand are randomly selected and squeezed until no more toothpaste will come out. Then each tube is cut open and the amount remaining is weighed, resulting in the following data (consistent with what the cited article reported):

\(.53,.65,.46,.50,.37.\)

Does it appear that the true average amount left is less than \(10\% \) of the advertised net contents?

a. Check the validity of any assumptions necessary for testing the appropriate hypotheses.

b. Carry out a test of the appropriate hypotheses using a significance level of \(.05\). Would your conclusion change if a significance level of \(.01\) had been used?

c. Describe in context type I and II errors, and say which error might have been made in reaching a conclusion.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.