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Chapter 7: Problem 6

What is a sampling distribution?

Short Answer

Expert verified
A sampling distribution is the distribution of a statistic (like the mean) from all possible samples of a population.

Step by step solution

01

Understanding Sampling

Sampling is the process of selecting a subset of individuals, data, or observations from a larger population in order to estimate characteristics of the whole population. The subset is called a sample.
02

Introduction to Sampling Distribution

A sampling distribution is the probability distribution of a given statistic (such as mean or proportion) that is calculated from a sample of data taken from a larger population. This distribution represents the variation of the statistic across different possible samples.
03

Calculating Different Sample Statistics

To understand sampling distribution, consider a population and repeatedly take samples of a certain size (each including a number of observations from the population) and calculate a statistic (such as the mean) for each sample.
04

Constructing the Sampling Distribution

The sampling distribution is constructed by plotting these calculated statistics (like sample means) from all possible samples of the same size. The shape of this distribution shows how sample means tend to distribute around the population mean.
05

Importance of the Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of sample means will tend to be normal (Gaussian) in shape as the sample size grows, regardless of the population's original distribution.

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Key Concepts

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

Sampling
Sampling is like selecting a small group from a bigger crowd, to help understand features of the whole group. Imagine wanting to know the average height of students in a school. Measuring every student can be tiresome and time-consuming. So, instead of picking everyone, you choose a few students randomly. This smaller group that you select is known as a "sample."
  • The process is called "sampling," which means choosing a sample from the entire population.
  • The main goal is to ensure this sample is representative of the broader population, allowing you to make accurate predictions about the population as a whole.
Picking a good sample is crucial. The way you select it determines the success of your predictions.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fascinating concept in statistics. It tells us that when you take multiple samples from a population and calculate their means, something magical happens.

No matter what shape the original data distribution has, the means of these samples will form a normal distribution if the sample size is big enough.
  • This happens even if the original data isn't normally distributed.
  • The bigger the sample size, the more the sample means resembled a bell-shaped curve.
This concept is crucial because it allows us to make predictions using the normal distribution, simplifying calculations and making statistical methods more reliable.
Probability Distribution
A probability distribution is like a map of all the possible outcomes of an experiment and how likely each outcome is to occur. It's a little guide that helps statisticians predict what might happen. In terms of sampling distribution,
  • it represents the range of values a sample statistic can take and their probabilities.
  • This lets us understand the variability of the statistic across different samples.
By knowing the probability distribution of a statistic, we can gauge its likelihood and ensure our interpretations are based on solid data.
Sample Statistics
Sample statistics are numbers that summarize the features of a sample. They include things like the sample mean, median, and variance. These statistics help us estimate the characteristics of the entire population.

When you calculate these statistics for different samples and plot them, you create a sampling distribution. This distribution helps illustrate how these statistics may vary.
  • Sample mean is one of the most commonly used sample statistics.
  • Other examples include sample standard deviation and range.
By using sample statistics wisely, you can get meaningful insights into the overall population even when working with just a sample.

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Most popular questions from this chapter

Suppose we have a binomial distribution with \(n\) trials and probability of success \(p\). The random variable \(r\) is the number of successes in the \(n\) trials, and the random variable representing the proportion of successes is \(\hat{p}=r / n\). (a) \(n=50 ; p=0.36\); Compute \(P(0.30 \leq \hat{p} \leq 0.45)\). (b) \(n=38 ; p=0.25 ;\) Compute the probability that \(\hat{p}\) will exceed \(0.35\). (c) \(n=41 ; p=0.09 ;\) Can we approximate \(\hat{p}\) by a normal distribution? Explain.

What is the standard deviation of a sampling distribution called?

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?

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.

Consider a binomial experiment with \(n\) trials for which \(n p>5\) and \(n q>5\). What is the value of the continuity correction when (a) \(n=25\) ? (b) \(n=100\) ? (c) As the value of \(n\) increases, does the continuity correction value increase, decrease, or stay the same?
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