Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 7: Problem 74

A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with mean \(3 \mathrm{~cm}\) and standard deviation \(0.1 \mathrm{~cm}\). The specifications call for corks with diameters between \(2.9\) and \(3.1 \mathrm{~cm}\). A cork not meeting the specifications is considered defective. (A cork that is too small leaks and causes the wine to deteriorate; a cork that is too large doesn't fit in the bottle.) What proportion of corks produced by this machine are defective?

Short Answer

Expert verified
The proportion of corks produced by this machine that are defective is approximately 31.73%.

Step by step solution

01

Calculate the Z-scores

The z-score represents how many standard deviations an element is from the mean. It can be calculated using the formula: \( z = (X - \mu) / \sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation. For this exercise two z-scores will be calculated, one for the lower specification limit (2.9 cm), and one for the upper specification limit (3.1 cm). Let \( z_{2.9} = (2.9 - 3) / 0.1 = -1 \) and \( z_{3.1} = (3.1 - 3) / 0.1 = 1 \). This means that all the corks with diameters between 2.9 cm to 3.1 cm are within one standard deviation on either side of the mean.
02

Calculate the Proportion of Good Corks

Once the z-scores are calculated, the next step is to find the proportion of corks that lie within the given specifications. The z-table or standard normal distribution table is used to find the probability that a randomly selected cork has diameter between 2.9 cm and 3.1 cm. According to the Z-table, the proportion of data within one standard deviation (-1 to 1) of the mean is approximately 0.6827 or 68.27%.
03

Calculate the Proportion of Defective Corks

To get the proportion of defective corks: a cork can either be good or defective, so the probabilities of these two must add up to 1. So, the proportion of defective corks is given by \( 1-0.6827 = 0.3173 \) or 31.73%

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 article on polygraph testing of FBI agents referenced in Exercise \(7.51\) indicated that the probability of a false-positive (a trustworthy person who nonetheless fails the test) is \(.15 .\) Let \(x\) be the number of trustworthy \(\mathrm{FBI}\) agents tested until someone fails the test. a. What is the probability distribution of \(x ?\) b. What is the probability that the first false-positive will occur when the third person is tested? c. What is the probability that fewer than four are tested before the first false-positive occurs? d. What is the probability that more than three agents are tested before the first false-positive occurs?

Suppose that your statistics professor tells you that the scores on a midterm exam were approximately normally distributed with a mean of 78 and a standard deviation of 7 . The top \(15 \%\) of all scores have been designated A's. Your score is \(89 .\) Did you receive an A? Explain.

Exercise \(7.8\) gave the following probability distribution for \(x=\) the number of courses for which a randomly selected student at a certain university is registered: $$ \begin{array}{lrrrrrrr} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ p(x) & .02 & .03 & .09 & .25 & .40 & .16 & .05 \end{array} $$ It can be easily verified that \(\mu=4.66\) and \(\sigma=1.20\). a. Because \(\mu-\sigma=3.46\), the \(x\) values 1,2 , and 3 are more than 1 standard deviation below the mean. What is the probability that \(x\) is more than 1 standard deviation below its mean? b. What \(x\) values are more than 2 standard deviations away from the mean value (i.e., either less than \(\mu-2 \sigma\) or greater than \(\mu+2 \sigma)\) ? What is the probability that \(x\) is more than 2 standard deviations away from its mean value?

Let \(z\) denote a variable that has a standard normal distribution. Determine the value \(z^{*}\) to satisfy the following conditions: a. \(P\left(zz^{*}\right)=.02\) e. \(P\left(z>z^{*}\right)=.01\) f. \(P\left(z>z^{*}\right.\) or \(\left.z<-z^{*}\right)=.20\)

The lightbulbs used to provide exterior lighting for a large office building have an average lifetime of \(700 \mathrm{hr}\). If length of life is approximately normally distributed with a standard deviation of \(50 \mathrm{hr}\), how often should all the bulbs be replaced so that no more than \(20 \%\) of the bulbs will have already burned out?7.122 The lightbulbs used to provide exterior lighting for a large office building have an average lifetime of \(700 \mathrm{hr}\). If length of life is approximately normally distributed with a standard deviation of \(50 \mathrm{hr}\), how often should all the bulbs be replaced so that no more than \(20 \%\) of the bulbs will have already burned out?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.

Sign-up for free