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Chapter 7: Problem 31

A population has a normal distribution. A sample of size \(n\) is selected from this population. Describe the shape of the sampling distribution of the sample mean for each of the following cases. a. \(n=23\) b. \(n=450\)

Short Answer

Expert verified
For both cases \(n=23\) and \(n=450\), the sampling distribution of the sample means will be normally distributed.

Step by step solution

01

Understanding the scenario

We are given a population that follows a normal distribution. As the population is normal, the sample mean will also follow a normal distribution regardless of sample size. For each case, we need to describe the shape of the sampling distribution of the mean.
02

Describe the sampling distribution for \(n=23\)

In this scenario, the sample size \(n=23\), which is a little bit less than 30. But as the source population is normally distributed, then the sample means will also be normally distributed. So, in this case, the shape of the sampling distribution will be normally distributed.
03

Describe the sampling distribution for \(n=450\)

In this scenario, the sample size \(n=450\) is greatly larger than 30. According to the Central Limit Theorem, the distribution of the sample means will be nearly normal, irrespective of the shape of the population distribution. But since the population was normal to begin with, then the sampling distribution of sample means will certainly be normal.

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Key Concepts

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

Normal Distribution
Normal distribution is a fundamental concept in statistics commonly known as the bell curve because of its symmetrical, bell-shaped appearance. It describes how data points are distributed around a central value, with most occurrences clustering around the mean and fewer instances towards the extremes.
  • In a normal distribution, the mean, median, and mode are all equal and located at the center.
  • The curve is symmetric about the mean.
  • The tails of the curve approach but never touch the horizontal axis.
  • 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
Understanding normal distribution is crucial because many statistical tests and methods assume that data follows this pattern. In sampling, if the population is normally distributed, the sampling distribution of the sample mean will also be normal.
Central Limit Theorem
The Central Limit Theorem (CLT) is a key principle in statistics that describes the characteristics of the sampling distribution of the sample mean. It states that as the sample size becomes large, the sampling distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution.
  • This theorem is powerful because it allows us to make inferences about the population mean using the sample mean.
  • CLT assures us that for a sufficiently large sample size, the sample mean's distribution will become approximately normal.
  • Usually, a sample size of 30 or more is considered sufficient for the CLT to hold true.
The CLT provides the foundation for many statistical procedures, making it essential for students to understand this concept when analyzing data from various types of populations.
Sample Size Effects
Sample size plays a crucial role in the accuracy and reliability of statistical conclusions. The larger the sample size, the more information it carries about the population, often leading to a better approximation of the population parameters.
  • Small sample sizes can lead to greater variability and less reliable estimates.
  • Larger samples tend to yield more precise estimates with narrower confidence intervals.
  • According to the CLT, larger samples result in a sampling distribution that is approximately normal.
Understanding the effect of sample size helps in designing studies and experiments, ensuring that the sample is adequately large enough to provide reliable and valid insights into the entire population.
Population Distribution
Population distribution refers to the way values of a variable are distributed among all possible instances in a dataset. It is a theoretical distribution that describes an entire dataset rather than just a sample.
  • The nature of the population distribution has implications for how samples from the population behave.
  • If the population is normally distributed, then any sample taken will also show similar characteristics.
  • If the population distribution is not normal, a larger sample size is needed to approximate the normal distribution, as described by the Central Limit Theorem.
Understanding the population distribution helps in deciding the appropriate statistical methods and in interpreting the results of data analysis. It serves as a backdrop against which statistical hypotheses and inferences are made.

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Most popular questions from this chapter

Consider a large population with \(\mu=90\) and \(\sigma=18\). Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample mean, \(\bar{x}\), for a sample size of a. 10 b. 35

Mong Corporation makes auto batteries. The company claims that \(80 \%\) of its LL70 batteries are good for 70 months or longer. Assume that this claim is true. Let \(\hat{p}\) be the proportion in a sample of 100 such batteries that are good for 70 months or longer. a. What is the probability that this sample proportion is within \(.05\) of the population proportion? b. What is the probability that this sample proportion is less than the population proportion by \(.06\) or more? c. What is the probability that this sample proportion is greater than the population proportion by 07 or more?

Beginning in the second half of 2011 , there were widespread protests in many American cities that were primarily against Wall Street corruption and the gap between the rich and the poor in America. According to a Time Magazine/ABT SRBI poll conducted by telephone during October \(9-10,2011,86 \%\) of adults who were familiar with those protests agreed that Wall Street and lobbyists have too much influence in Washington (The New York Times, October 22, 2011). Assume that this percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 400 American adults who hold the opinion that Wall Street and lobbyists have too much influence in Washington. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

The amounts of electricity bills for all households in a particular city have an approximately normal distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30 .\) Let \(\bar{x}\) be the mean amount of electricity bills for a random sample of 25 households selected from this city. Find the mean and standard deviation of \(\bar{x}\), and comment on the shape of its sampling distribution.

A population of \(N=5000\) has \(\sigma=25 .\) In each of the following cases, which formula will you use to calculate \(\sigma_{\bar{x}}\) and why? Using the appropriate formula, calculate \(\sigma_{\bar{x}}\) for each of these cases. a. \(n=300\) b. \(n=100\)
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