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Chapter 7: Problem 1

What is the standard error of a sampling distribution?

Short Answer

Expert verified
The standard error quantifies the dispersion of a sample statistic within the sampling distribution.

Step by step solution

01

Understand the Concept of Standard Error

The standard error (SE) of a sampling distribution measures the variability or dispersion of a sample statistic, such as the sample mean, from the true population parameter. It indicates how much the sample statistic is expected to fluctuate due to random sampling variation.
02

Identify the Formula for Standard Error

The standard error of the mean can be calculated using the formula: \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size.
03

Recognize When to Use Standard Error

Use the standard error when you need to estimate how far a sample mean is likely to be from the population mean. It is particularly useful when making inferences about the population from a sample.

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Key Concepts

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

Understanding Sampling Distribution
A sampling distribution is a crucial concept in statistics and helps us understand how means of samples behave. When you repeatedly take samples from a population and calculate the sample statistic, such as the mean, you can create a distribution of those sample statistics. This is what we call the sampling distribution.
It helps in estimating the reliability of a statistical measure because it considers how sample statistics vary.
This variation comes due to the different possible samples you can draw from a population.
  • It represents how a sample mean or other statistic would behave if we took many samples.
  • The center of this distribution is usually close to the parameter of the population.
Knowing about sampling distribution enables us to apply the standard error effectively to make inferences about the population from which the sample is drawn.
Defining Population Parameter
A population parameter is a value that represents a particular characteristic of the entire population, such as the mean or standard deviation. Unlike a sample statistic, a population parameter is a fixed value if you could measure the entire population.
Common examples of population parameters include:
  • Population Mean (\( \mu \))
  • Population Standard Deviation (\( \sigma \))
  • Population Proportion
However, we rarely know these values exactly because they require data from every individual in the population, which is often impractical. That鈥檚 why sample statistics and concepts like standard error are vital鈥攖hey help us estimate these population parameters based on samples.
Explaining Random Sampling Variation
Random sampling variation refers to the differences that naturally occur between sample statistics and the true population parameters. When you select different samples from the same population, each sample's composition may lead to slight variations in the calculated statistics.
These variations originate from the randomness inherent in the sampling process.
  • The degree of random sampling variation impacts the standard error.
  • Higher variability in sampling means a larger standard error, indicating less reliability in the sample statistic as a measure of the population parameter.
  • The larger the sample size, the smaller the impact of random sampling variation.
Understanding and accounting for random sampling variation is essential for making accurate conclusions from sample data.
Clarifying Sample Statistic
A sample statistic is a figure that summarizes or describes an aspect of a sample drawn from a population, such as the sample mean or sample standard deviation. These statistics provide an estimate of the population parameters. Since we often cannot reach every member of a population, we use these sample statistics to gain insights.
  • Sample Mean (\( \bar{x} \))
  • Sample Standard Deviation (s)
  • Sample Proportion
The role of sample statistics is crucial as they form the basis for inferential statistics, allowing researchers to make predictions or assumptions about a population. An understanding of how these values relate to population parameters and their variability helps us interpret statistical analyses accurately.

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Most popular questions from this chapter

What is a random sample from a population? (Hint: See Section 1.2.)

Consider two \(\bar{x}\) distributions corresponding to the same \(x\) distribution. The first \(\bar{x}\) distribution is based on samples of size \(n=100\) and the second is based on samples of size \(n=225 .\) Which \(\bar{x}\) distribution has the smaller standard error? Explain.

P-Chart: Aluminum Cans A high-speed metal stamp machine produces \(12 .\) ounce aluminum beverage cans. The cans are mass produced in lots of 110 cans for each square sheet of aluminum fed into the machine. However, some of the cans come out of the die stamp with folds and wrinkles. These are defective cans that must be recycled. Let us view each can as a binomial trial, where success is defined to mean the can is defective. So we have \(n=110\) trials (cans), and the random variable \(r\) is the number of defective cans. A test run of 15 consecutive aluminum sheets gave the following numbers \(r\) of defective cans. $$ \begin{array}{l|rrrrrrrr} \hline \text { Test sheet } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline r & 8 & 11 & 6 & 9 & 12 & 8 & 7 & 11 \\ \hline \hat{p}=r / 110 & 0.07 & 0.10 & 0.05 & 0.08 & 0.11 & 0.07 & 0.06 & 0.10 \\\ \hline \text { Test sheet } & 9 & 10 & 11 & 12 & 13 & 14 & 15 & \\ \hline r & 10 & 7 & 9 & 6 & 12 & 7 & 10 & \\ \hline \hat{p}=r / 110 & 0.09 & 0.06 & 0.08 & 0.05 & 0.11 & 0.06 & 0.09 & \\ \hline \end{array} $$ Make a \(P\) -Chart and list any out-of-control signals by type (I, II, or III). Does it appear from the sequential test runs that the production process is in reasonable control? Explain.

Focus Problem: Impulse Buying Let \(x\) represent the dollar amount spent on supermarket impulse buying in a 10 -minute (unplanned) shopping interval. Based on a Denver Post article, the mean of the \(x\) distribution is about \(\$ 20\) and the estimated standard deviation is about \(\$ 7\). (a) Consider a random sample of \(n=100\) customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of \(\bar{x}\), the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the \(\bar{x}\) distribution? Is it necessary to make any assumption about the \(x\) distribution? Explain. (b) What is the probability that \(\bar{x}\) is between \(\$ 18\) and \(\$ 22 ?\) (c) Let us assume that \(x\) has a distribution that is approximately normal. What is the probability that \(x\) is between \(\$ 18\) and \(\$ 22 ?\) (d) Interpretation: In part (b), we used \(\bar{x}\), the average amount spent, computed for 100 customers. In part (c), we used \(x\), the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen? In this example, \(\bar{x}\) is a much more predictable or reliable statistic than \(x\). Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

(a) If we have a distribution of \(x\) values that is more or less mound-shaped and somewhat symmetrical, what is the sample size needed to claim that the distribution of sample means \(\bar{x}\) from random samples of that size is approximately normal? (b) If the original distribution of \(x\) values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means \(\bar{x}\) taken from random samples of a given size is normal?
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