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Chapter 6: Problem 4

Annual incomes are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a large \((n>30)\) random sample of annual incomes. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is large? Why or why not?

Short Answer

Expert verified
Yes, the distribution of the sample mean can be approximated by a normal distribution for a large sample size due to the Central Limit Theorem.

Step by step solution

01

- Understanding the Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will tend to be approximately normally distributed, provided the sample size is sufficiently large (typically n > 30), regardless of the population distribution.
02

- Applying the Central Limit Theorem

Since the sample size is large (n > 30), the CLT suggests that the distribution of the sample mean of the annual incomes will be approximately normal, even though the population distribution is skewed to the right.
03

- Conclusion

Thus, due to the CLT, we can approximate the distribution of the sample mean of annual incomes by a normal distribution for a large sample size, despite the original skewness of the population distribution.

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Key Concepts

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

Sampling Distribution
When we talk about a 'sampling distribution', we refer to the probability distribution of a certain statistic based on many samples from the same population. In this exercise, our statistic is the sample mean of annual incomes. Each sample mean we calculate from many samples creates a sampling distribution of the sample mean.

Even if the original population distribution is skewed to the right, the Central Limit Theorem (CLT) helps us understand that the sampling distribution of the sample mean will be approximately normal, provided we have a sufficiently large sample size. This means that when we create a histogram of the sample means, it will resemble a bell-shaped curve, even if the original population does not.

In simpler terms, while individual incomes might vary and be skewed, the average of these incomes from different samples will tend to follow a normal pattern.
Normal Distribution
The normal distribution, often called the bell curve, is a pattern where data points are most frequently found around the mean and less frequently as you move away from the mean. It looks like a symmetric bell.

Even though the original distribution of annual incomes is skewed to the right, we can use the Central Limit Theorem to approximate the sampling distribution of the sample mean as normal, given a large sample size. This shift happens because the average values of the samples will balance out extremes or skewness present in the original distribution.

This concept is extremely useful because many statistical tests and procedures assume a normal distribution. By using the CLT, we can apply these tests to sample means even if the underlying population is not normally distributed.

Understanding the normal distribution helps us apply these statistical methods more accurately and with confidence, knowing that the average of large samples leans towards normality.
Sample Size
Sample size plays a crucial role in applying the Central Limit Theorem. According to the theorem, a larger sample size (typically n > 30) helps the sampling distribution of the sample mean approach a normal distribution.

In our exercise, the phrase 'n>30' exemplifies this aspect. With a large sample size, individual variations, and skewness in the data are smoothed out, making the sample mean more normally distributed.

This is why statisticians emphasize collecting sufficient data points. A small sample might not capture the population's characteristics correctly, leading to erroneous conclusions. On the other hand, with a large enough sample size, we can be more confident that our analysis and inferences drawn from the sample mean will be accurate and reliable.

So, when working with skewed populations but needing to use techniques and tests that assume normality, ensure your sample size is large enough!

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Most popular questions from this chapter

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According to the website www.torchmate.com, "manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter." Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of \(18.2\) in. and a standard deviation of \(1.0\) in. (based on data from the National Health and Nutrition Examination Survey). a. What percentage of men will fit into the manhole? b. Assume that the Connecticut's Eversource company employs 36 men who work in manholes. If 36 men are randomly selected, what is the probability that their mean shoulder breadth is less than \(18.5\) in.? Does this result suggest that money can be saved by making smaller manholes with a diameter of \(18.5 \mathrm{in} . ?\) Why or why not?
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