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Chapter 6: Problem 2

What is the standard deviation of a sampling distribution called?

Short Answer

Expert verified
The standard deviation of a sampling distribution is called the standard error.

Step by step solution

01

Understanding the Question

The question is asking for a specific term used in statistics that refers to the standard deviation of a sampling distribution. Let's explore the concept of sampling distribution to identify the term.
02

Define Sampling Distribution

A sampling distribution is the probability distribution of a given statistic, such as the sample mean, calculated from a large number of samples drawn from a specific population.
03

Introduce the Term

The standard deviation of the sampling distribution of a statistic, most commonly the sample mean, is referred to by a specific term. This term represents the typical error or variation expected in the sampled statistic, due to random sampling.
04

Identify the Term

The term used to describe the standard deviation of a sampling distribution of the sample mean (or any statistic) is the "standard error." Specifically, when referring to the sample mean, it is called the standard error of the mean.

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Key Concepts

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

Sampling Distribution
A sampling distribution can be thought of as the distribution of all possible samples of a particular statistic from a population. Imagine repeatedly taking random samples from a population and calculating the sample mean for each one.
This collection of sample means forms the sampling distribution.

In practice:
  • A sampling distribution is used to examine the variability of a statistic.
  • It helps in making inferences about population parameters based on sample statistics.
  • The shape of a sampling distribution will often resemble a normal distribution with a large enough sample size due to the Central Limit Theorem.
This is crucial because it allows statisticians to quantify the uncertainty of a statistic, like the sample mean, through measures such as the standard error.
Standard Deviation
The standard deviation is a measure used to describe the spread or dispersion of a set of values. In the context of a sampling distribution, it tells us how much the sample statistics differ from the true population parameter.

Here's why it matters:
  • A smaller standard deviation indicates that the sample values are closely clustered around the mean, suggesting less variability and more reliability.
  • A larger standard deviation signifies more spread and variability around the mean, pointing towards a broader range of possible values.
  • In a sampling distribution, the standard deviation is specifically important because it quantifies the expected variation from one sample to another, due to random sampling.
This is essential for determining the precision of sample estimates and helps in error measurement, primarily leading to a better understanding of the standard error.
Sample Mean
The sample mean is one of the most common statistics calculated from a sample and represents the average of the data points within that sample. It serves as an estimator for the population mean.

Understanding the sample mean:
  • The sample mean is calculated by adding all sample values together, then dividing by the number of values.
  • It provides a single summary statistic for the entire sample, simplifying analysis and interpretation.
  • When repeated sampling is done, the collection of these sample means forms a sampling distribution of the mean.
The sample mean's importance lies in its use for estimating population parameters, and it's often central to statistical inference and analysis. Because of this, understanding its spread or variation, quantified by the standard error, is vital for accurate results.

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Most popular questions from this chapter

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(8 \leq x \leq 12) ; \mu=15 ; \sigma=3.2 $$

Templeton World is a mutual fund that invests in both U.S. and foreign markets. Let \(x\) be a random variable that represents the monthly percentage return for the Templeton World fund. Based on information from the Morningstar Guide to Mutual Funds (available in most libraries), \(x\) has mean \(\mu=1.6 \%\) and standard deviation \(\sigma=0.9 \%\). (a) Templeton World fund has over 250 stocks that combine together to give the overall monthly percentage return \(x\). We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return \(x\) for Templeton World fund is itself an average return computed using all 250 stocks in the fund. Why would this indicate that \(x\) has an approximately normal distribution? Explain. Hint: See the discussion after Theorem \(7.2\). (b) After 6 months, what is the probability that the average monthly percentage return \(\bar{x}\) will be between \(1 \%\) and \(2 \%\) ? Hint: See Theorem \(7.1\), and assume that \(x\) has a normal distribution as based on part (a). (c) After 2 years, what is the probability that \(\bar{x}\) will be between \(1 \%\) and \(2 \%\) ? (d) Compare your answers to parts (b) and (c). Did the probability increase as \(n\) (number of months) increased? Why would this happen? (e) If after 2 years the average monthly percentage return \(\bar{x}\) was less than \(1 \%\), would that tend to shake your confidence in the statement that \(\mu=1.6 \%\) ? Might you suspect that \(\mu\) has slipped below \(1.6 \%\) ? Explain.

Give an example of a specific sampling distribution we studied in this section. Outline other possible examples of sampling distributions from areas such as business administration, economics, finance, psychology, political science, sociology, biology, medical science, sports, engineering, chemistry, linguistics, and so on.

Suppose \(x\) has a distribution with a mean of 8 and a standard deviation of \(16 .\) Random samples of size \(n=64\) are drawn. (a) Describe the \(\bar{x}\) distribution and compute the mean and standard deviation of the distribution. (b) Find the \(z\) value corresponding to \(\bar{x}=9\). (c) Find \(P(\bar{x}>9)\). (d) Would it be unusual for a random sample of size 64 from the \(x\) distribution to have a sample mean greater than 9? Explain.

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. Between \(z=-2.42\) and \(z=-1.77\)
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