Book cover for Probability and Statistics for Engineers and Scientists

Probability and Statistics for Engineers and Scientists

Ronald E. Walpole, Raymond H. Myers, Sharon L

ISBN #9780131877115

8th Edition

1,256 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the foundational principles of statistical inference, highlighting the pivotal role of sampling distributions in linking sample statistics to population parameters. It underscores the importance of the central limit theorem in understanding the behavior of the sample mean and details estimation techniques for one-sample and two-sample scenarios. The chapter sets the stage for more formal statistical inference methodology, emphasizing unbiased estimation and the necessity of understanding variability in sample-based conclusions.

Learning Objectives

1

Explain the concept of sampling distributions and how they form the basis of statistical inference.

2

Describe the role of the central limit theorem in understanding the distribution of the sample mean.

3

Outline the process of estimating population parameters using sample statistics.

4

Differentiate between estimation procedures for one-sample and two-sample problems.

Key Concepts

CONCEPT

DEFINITION

Sampling Distribution

The probability distribution of a given statistic (like the sample mean or variance) based on a random sample from a population.

Sample Mean (XÌ„)

The average of the observations in a sample, used as an estimator for the population mean.

Sample Variance (S²)

A measure of the dispersion of sample data points around the sample mean, used to estimate the population variance.

Central Limit Theorem (CLT)

A statistical theory that states that, for a large sample size, the sampling distribution of the sample mean will be approximately normally distributed regardless of the original population distribution.

Statistical Inference

The process of using data from a sample to make estimates, decisions, or predictions about a population.

Estimation

A method used in statistics to infer the value of an unknown parameter in a population based on sample data.

Example Problems

Example 1

Let us define $\mathrm{A}^{\prime 2}=\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} / n$ Show that$$E\left(S^{\prime}\right)=[(n-L) / n] \sigma$$ and hence $S^{\prime}$ ' is a biased estimator for $\sigma^{-2}$.

Example 2

If $\mathrm{A}^{\prime}$ is a binomial random variable, show that (a) $P=X / n$ is an unbiased estimator of $p ;$ (b) $P^{\prime}=\frac{X+\sqrt{n} / 2}{n+w}$ is a biased estimator of $p$.

Example 3

Show that the estimator $P^{\prime}$ of Exercise $9.2(\mathrm{~b})$ becomes unbiased as $n \rightarrow>00 .$

Example 4

An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a $96 \%$ confidence interval for the population mean of all bulbs produced by this firm.

Example 5

Many cardiac patients wear implanted pacemakers to control their heartbeat. A plastic connector module mounts on the top of the pacemaker. Assuming a standard deviation of 0.0015 and an approximate normal distribution, find a $95 \%$ confidence interval for the mean of all connector modules made by a certain manufacturing company. A random sample of 75 modules has an average of 0.310 inch.
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Step-by-Step Explanations

QUESTION

How can we estimate the population mean using the sample mean and its sampling distribution?

STEP-BY-STEP ANSWER:

Step 1: Identify the sample mean (XÌ„) from the collected data.
Step 2: Understand that XÌ„ is a statistic from the sampling distribution, which, according to the Central Limit Theorem, approximates a normal distribution for large sample sizes.
Step 3: Recognize that the center of the sampling distribution is the population mean (μ). Thus, X̄ is an unbiased estimator of μ.
Step 4: Based on these properties, conclude that the sample mean is used as the best point estimate of the population mean.
Final Answer: The sample mean X̄ is used to estimate the population mean μ because its sampling distribution centers at μ, making it an unbiased and reliable estimator.

Estimating the Population Mean

QUESTION

Why is knowledge of the sampling distribution important when making inferences about population parameters?

STEP-BY-STEP ANSWER:

Step 1: Recognize that any statistic (e.g., sample mean or variance) has a probability distribution when considering all possible samples from the population.
Step 2: Understand that this distribution provides information on the variability and reliability of the estimator.
Step 3: Use the properties of the sampling distribution, such as its mean and variance, to construct confidence intervals and perform hypothesis testing.
Step 4: Realize that without this information, conclusions about the population parameters could be misleading or imprecise.
Final Answer: Knowledge of the sampling distribution is crucial because it underpins the methods for estimating population parameters, determining estimator reliability, and conducting valid hypothesis tests.

Understanding the Role of Sampling Distributions in Inferential Statistics

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Common Mistakes

  • Confusing the sample statistic with the population parameter, leading to overconfidence in estimates.
  • Ignoring the requirements of the central limit theorem, such as sufficiently large sample sizes, when applying it.
  • Failing to account for the variability inherent in the sampling distribution, which can result in misinterpreting the precision of an estimate.