/*! 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} Problem 50 Making auto parts A grinding mac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 50

Making auto parts A grinding machine in an auto parts plant prepares axles with a target diameter \(\mu=40.125\) millimeters \((\mathrm{mm}) .\) The machine has some variability, so the standard deviation of the diameters is \(\sigma=0.002 \mathrm{mm}\) . The machine operator inspects a random sample of 4 axles each hour for quality control purposes and records the sample mean diameter \(\overline{x} .\) Assuming that the process is working properly, what are the mean and standard deviation of the sampling distribution of \(\overline{x} ?\) Explain.

Short Answer

Expert verified
Mean: 40.125 mm, Standard Deviation of Sampling Distribution: 0.001 mm.

Step by step solution

01

Understanding the Problem

The problem involves a grinding machine that prepares axles with a specified target diameter and variability. We want to find the mean and standard deviation of the sampling distribution of the sample mean given a sample size and known population parameters.
02

Identify Known Values

From the problem statement, the known values are: the population mean \(\mu = 40.125\, \text{mm}\), the population standard deviation \(\sigma = 0.002\, \text{mm}\), and the sample size \(n = 4\).
03

Calculate the Mean of the Sampling Distribution

The mean of the sampling distribution of the sample mean \(\overline{x}\) is the same as the population mean. Hence, the mean of the sampling distribution is \(\mu_{\overline{x}} = \mu = 40.125\, \text{mm}\).
04

Determine the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution (also known as the standard error) is calculated using the formula: \(\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}\). Substituting the known values, we get \(\sigma_{\overline{x}} = \frac{0.002}{\sqrt{4}} = 0.001\, \text{mm}\).

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 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Sampling Error
In the world of statistics, sampling error is the difference between a sample measurement and the actual population parameter. Simply put, it's the error that arises when we use a sample to estimate something about the whole population.

Sampling error is inevitable whenever a sample, rather than the entire population, is used for analysis. It's inherent to the process of sampling itself. However, the goal is to minimize this error to get a more accurate estimation of the population.
  • Factors affecting sampling error include the sample size and the variability among the population individuals.
  • Increasing the sample size generally reduces the sampling error since larger samples tend to provide more reliable estimates.
  • Understanding and calculating sampling error helps in determining the reliability and accuracy of the sample results.
These insights are crucial, especially in quality control settings, like monitoring machine operations, where precision matters a lot.
Exploring Standard Deviation
Standard deviation is a statistical term that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider range of values.

In our scenario with the grinding machine, the standard deviation of 0.002 mm represents the typical variation in axle diameters from their target size.
  • The formula for standard deviation in a population is \( \sigma \).
  • For a sample, this is labeled as \( s \), but in this problem involving sampling distribution, we are primarily concerned with \( \sigma \).
Standard deviation is pivotal, as it's used to compute the standard error. This is a key factor when evaluating the precision of the sample mean, helping ensure the machine's outputs are within the allowable variance to meet quality standards.
Defining Sample Mean
The sample mean, denoted as \( \overline{x} \), is the average of the measurements in a sample. It serves as an estimate of the population mean. In our exercise, the sample mean diameter represents the average diameter of the 4 axles inspected each hour.
  • To compute the sample mean, you add up all the sample measurements and divide by the number of measurements \( n \).
  • The sample mean helps monitor and control the machine鈥檚 performance by comparing it with the target population mean \( \mu \).
  • When sampling is done correctly, the sample mean is an unbiased and consistent estimator of the population mean.
The precision of the sample mean's estimate is critical in a manufacturing environment, as it determines if machine adjustments are necessary to maintain product quality.

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

For each boldface number in Exercises 5 to \(8,(1)\) state whether it is a parameter or a statistic and (2) use appropriate notation to describe each number; for example, \(p=0.65\) Get your bearings A large container of ball bearings has mean diameter 2.5003 centimeters \((\mathrm{cm}) .\) This is within the specifications for acceptance of the container by the purchaser. By chance, an inspector chooses 100 bearings from the container that have mean diameter 2.5009 \(\mathrm{cm} .\) Because this is outside the specified limits, the container is mistakenly rejected.

Multiple choice: Select the best answer for Exercises 43 to \(46,\) which refer to the following setting. The magazine Sports Illustrated asked a random sample of 750 Division I college athletes, "Do you believe performance- enhancing drugs are a problem in college sports?" Suppose that 30\(\%\) of all Division I athletes think that these drugs are a problem. Let \(\hat{p}\) be the sample proportion who say that these drugs are a problem. The sampling distribution of \(\hat{p}\) has mean \(\begin{array}{ll}{\text { (a) } 225 .} & {\text { (c) } 0.017 \text { . (e) none of these. }} \\ {\text { (b) } 0.30 .} & {\text { (d) } 0}\end{array}\)

Bottling cola A bottling company uses a flling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliters \((\mathrm{ml}) .\) In fact, the contents vary according to a Normal distribution with mean \(\mu=298 \mathrm{ml}\) and standard deviation \(\sigma=3 \mathrm{ml}\) (a) What is the probability that an individual bottle contains less than 295 \(\mathrm{ml}\) ? Show your work. (b) What is the probability that the mean contents of six randomly selected bottles is less than 295 \(\mathrm{ml}\) ? Show your work.

Airport security The Transportation Security Administration \((\mathrm{TSA})\) is responsible for airport safety. On some flights, TSA officers randomly select passengers for an extra security check before boarding. One such flight had 76 passengers \(-12\) in first class and 64 in coach class. TSA officers selected an SRS of 10 passengers for screening. Let \(\hat{p}\) be the proportion of first-class passengers in the sample. (a) Is the 10\(\%\) condition met in this case? Justify your answer. (b) Is the Normal condition met in this case? Justify your answer.

Predict the election Just before a presidential election, a national opinion poll increases the size of its weekly random sample from the usual 1500 people to 4000 people. (a) Does the larger random sample reduce the bias of the poll result? Explain. (b) Does it reduce the variability of the result? Explain.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.