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Chapter 17: Problem 33

A sample is chosen randomly from a population that can be described by a Normal model. a) What's the sampling distribution model for the sample mean? Describe shape, center, and spread. b) If we choose a larger sample, what's the effect on this sampling distribution model?

Short Answer

Expert verified
The sampling distribution model is Normal with mean \(\mu\) and standard error \(\frac{\sigma}{\sqrt{n}}\). A larger sample size reduces the standard error, making the distribution narrower.

Step by step solution

01

Identify Sampling Distribution Model

When a sample is randomly chosen from a population described by a Normal model, the sampling distribution of the sample mean can also be modeled by a Normal distribution. This is due to the Central Limit Theorem, which states that the distribution of the sample means will be approximately Normal if the sample size is sufficiently large.
02

Describe the Shape of the Sampling Distribution

The shape of the sampling distribution of the sample mean is Normal. This is because the population is already Normal, and any sample mean from this population will also produce a Normal distribution in the limit.
03

Determine the Center of the Sampling Distribution

The center of the sampling distribution is at the population mean, \(\mu\). This means that the mean of all possible sample means will equal the mean of the population from which they were drawn.
04

Calculate the Spread of the Sampling Distribution

The spread of the sampling distribution is quantified by the standard deviation of the sample mean, also known as the standard error. It is calculated as \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the standard deviation of the population, and \(n\) is the sample size. This formula shows that the standard error decreases as the sample size increases.
05

Discuss the Effect of a Larger Sample Size

Increasing the sample size affects the sampling distribution by reducing the standard error \(\frac{\sigma}{\sqrt{n}}\), thus making the distribution narrower. A larger sample size results in a sampling distribution that is more tightly clustered around the population mean, increasing the precision of the sample mean.

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Key Concepts

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

Sampling Distribution
The concept of a "sampling distribution" describes the distribution of a statistic, like the sample mean, across many samples taken from the same population. Imagine you took multiple samples from a population and calculated the mean of each sample. The sampling distribution is essentially a collection of all these means.

What makes this useful is that even if the original population is not normally distributed, the sampling distribution of the mean will be normally distributed if the sample size is large enough, thanks to the Central Limit Theorem.
  • This makes it easier for statisticians to make inferences about the population mean based on the sample mean.
  • It ensures that the sampling distribution will tend to stabilize around the actual population parameter as more samples are taken.
Sample Mean
The "sample mean" is the average of all data points collected in a sample. It is a key statistic used to estimate the mean of the entire population. The importance of the sample mean lies in its ability to provide insights into the population as a whole.

Under the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normally distributed, even if the original population distribution is not perfectly normal, as long as the sample size is sufficiently large. This means that the sample mean is bound to conform to a shape that is predictable and mathematically manageable.
  • This allows for straightforward comparison with the population mean.
  • Its proximity to the actual population mean will generally improve with larger sample sizes due to reduced variance.
Normal Distribution
A "Normal distribution" or bell curve is a continuous probability distribution characterized by its symmetric shape. In a Normal distribution, most data points cluster around a central point (the mean), with symmetrical tails on either side.

This distribution is pivotal in statistics due to its natural occurrence and its properties, which allow various methods and tools to analyze data efficiently.
  • The Central Limit Theorem shows that the sampling distribution of the sample mean will be approximately normal for a large enough sample size, even if the population distribution is not.
  • This makes predictions and hypothesis testing simpler as researchers often assume a normal distribution for the sampling distribution model.
Standard Error
The "standard error" is the standard deviation of the sampling distribution of a statistic, commonly the sample mean. It's a measure of how much the sample mean is expected to vary from the actual population mean if we were to take different samples from the same population repeatedly.

The formula for the standard error is given by \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the standard deviation of the population, and \( n \) is the sample size. This formula reveals a couple of important insights:
  • The larger the sample size \( n \), the smaller the standard error. This means larger samples lead to more precise estimates of the population mean.
  • The reduction in standard error with increased sample size explains why using larger samples is often more effective in gaining accurate inferences about the population.

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Most popular questions from this chapter

The candy company claims that \(16 \%\) of the Milk Chocolate M\&M's it produces are green. Suppose that the candies are thoroughly mixed and then packaged in small bags containing about \(50 \mathrm{M} \& \mathrm{M}^{\prime} \mathrm{s}\). A class of elementary school students learning about percents opens several bags, counts the various colors of the candies, and calculates the proportion that are green. a) If we plot a histogram showing the proportions of green candies in the various bags, what shape would you expect it to have? b) Can that histogram be approximated by a Normal model? Explain. c) Where should the center of the histogram be? d) What should the standard deviation of the sampling distribution be?

Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days. a) What percentage of pregnancies should last between 270 and 280 days? b) At least how many days should the longest \(25 \%\) of all pregnancies last? c) Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let \(\bar{y}\) represent the mean length of their pregnancies. According to the Central Limit Theorem, what's the distribution of this sample mean, \(\bar{y}\) ? Specify the model, mean, and standard deviation. d) What's the probability that the mean duration of these patients' pregnancies will be less than 260 days?

The weight of potato chips in a medium size bag is stated to be 10 ounces. The amount that the packaging machine puts in these bags is believed to have a Normal model with mean 10.2 ounces and standard deviation 0.12 ounces. a) What fraction of all bags sold are underweight? b) Some of the chips are sold in "bargain packs" of 3 bags. What's the probability that none of the 3 is underweight? c) What's the probability that the mean weight of the 3 bags is below the stated amount? d) What's the probability that the mean weight of a 24 -bag case of potato chips is below 10 ounces?

According to a 2013 poll from Public Policy Polling, \(4 \%\) of American voters believe that shape-shifting reptilian people control our world by taking on human form and gaining power. Yes, you read that correctly! (This was a poll about conspiracy theories.) Assume that's the actual proportion of Americans who hold that belief. a) Use a binomial model to calculate the probability that, in a random sample of 100 people, at least \(6 \%\) of those in the sample believe the thing about reptilian people controlling our world. b) Use a Normal model to calculate the same probability. How does this compare with the answer in part a? c) That same poll found that \(51 \%\) of American voters believe there was a larger conspiracy responsible for the assassination of President Kennedy. Use a binomial model to calculate the probability that, in a random sample of 100 people, at least \(57 \%\) of those in the sample believe in the JFK conspiracy theory. d) Use a normal model to calculate the same probability. How does this compare with the answer in part c? c) What do these answers tell you about the importance of checking that \(n p\) and \(n q\) are both at least \(10 ?\)

In Chapter 5 we saw the distribution of the total compensation of the chief executive officers (CEOs) of the 800 largest U.S. companies (the Fortune 800 ). The average compensation (in thousands of dollars) is 10,307.31 and the standard deviation is 17,964.62 Here is a histogram of their annual compensations (in \(\$ 1000)\): a) Describe the histogram of Total Compensation. A research organization simulated sample means by drawing samples of \(30,50,100,\) and \(200,\) with replacement, from the 800 CEOs. The histograms show the distributions of means for many samples of each size. b) Explain how these histograms demonstrate what the Central Limit Theorem says about the sampling distribution model for sample means. Be sure to talk about shape, center, and spread. c) Comment on an oft-cited "rule of thumb" that "With a sample size of at least \(30,\) the sampling distribution of the mean is Normal"
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