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Chapter 6: Problem 7

(a) If we have a distribution of \(x\) values that is more or less mound-shaped and somewhat symmetrical, what is the sample size needed to claim that the distribution of sample means \(\bar{x}\) from random samples of that size is approximately normal? (b) If the original distribution of \(x\) values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means \(\bar{x}\) taken from random samples of a given size is normal?

Short Answer

Expert verified
(a) Sample size of 30 or more is needed. (b) No restrictions on sample size if the distribution is normal.

Step by step solution

01

Understanding Distribution Shape

When a distribution of data is described as mound-shaped and symmetrical, it suggests it follows a bell curve pattern similar to a normal distribution. The Central Limit Theorem (CLT) provides insights about sample means from such distributions.
02

Applying Central Limit Theorem

According to the Central Limit Theorem, if you take sufficiently large random samples from almost any distribution and calculate their means, the distribution of these sample means will be approximately normal. Generally, a sample size of at least 30 is considered sufficient.
03

Evaluating Original Normal Distribution

If the original distribution of the variable x is normal, then any sample size, regardless of how small, will result in a distribution of sample means that is also normal. This is due to the properties of normal distributions.
04

Determining Sample Size Requirements

For part (a), to claim that the distribution of sample means is approximately normal, a sample size of 30 or more is needed. For part (b), if the original distribution is normal, no restriction on sample size is necessary to ensure the normality of the distribution of sample means.

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Key Concepts

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

Sample Size and the Central Limit Theorem
Sample size is a key concept when applying the Central Limit Theorem (CLT). Imagine you have a population with any shape of distribution, perhaps slightly mound-shaped or even a bit skewed. When you draw random samples from this population and calculate the mean of each sample, CLT tells us that the distribution of these sample means will resemble a normal distribution, provided the samples are of adequate size.
How large should these samples be? Generally, a sample size of 30 or more is often deemed sufficient for the CLT to apply. This magic number is large enough to smooth out weird shapes in the original data and make the distribution of sample means normal. Of course, in practice, this number can vary based on how different the original distribution is from normal.
  • A sample size of at least 30 is commonly used.
  • Larger sample sizes help to achieve a normal distribution of sample means faster.
  • If the underlying distribution is extremely skewed, you may need a larger sample size.
Understanding Normal Distribution
The normal distribution is a cornerstone of statistics. It is often represented as the bell curve due to its distinctive shape. In this distribution, most data points are concentrated around the mean, with the probabilities of occurrences tailing off symmetrically as you move away in either direction.
Why is this shape so important? Well, it has unique properties, like being completely defined by its mean and standard deviation. This consistency makes the normal distribution extremely useful for statistical inference.
  • The mean, median, and mode are all the same in a normal distribution.
  • 68% of the data falls within one standard deviation of the mean.
  • 95% of it falls within two, and 99.7% falls within three standard deviations.
Distribution of Sample Means and Its Normality
When discussing the distribution of sample means, also known as the sampling distribution, the effect of the Central Limit Theorem becomes crucial. It states that regardless of the shape of the original population distribution, the sampling distribution of the mean will approximate a normal distribution as the sample size becomes larger. This holds true generally when the sample size reaches or exceeds 30.
However, if the population distribution itself is normal, then every sample mean drawn from this population will also create a normal distribution of sample means, regardless of the size. This is because the inherent nature of the normal distribution allows it to maintain its properties through sampling. Therefore, all you need is a valid (even small) sample to replicate the normal distribution properties.
  • The CLT helps us predict the shape of the distribution of sample means.
  • For non-normal populations, a larger sample size ensures the normality of sample means.
  • For normal populations, sample means are normally distributed regardless of sample size.

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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. $$ P(8 \leq x \leq 12) ; \mu=15 ; \sigma=3.2 $$

A relay microchip in a telecommunications satellite has a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of \(3.7\) months. When this computer-relay microchip malfunctions, the entire satellite is useless. A large London insurance company is going to insure the satellite for \(\$ 50\) million. Assume that the only part of the satellite in question is the microchip. All other components will work indefinitely. (a) For how many months should the satellite be insured to be \(99 \%\) confident that it will last beyond the insurance date? (b) If the satellite is insured for 84 months, what is the probability that it will malfunction before the insurance coverage ends? (c) If the satellite is insured for 84 months, what is the expected loss to the insurance company? (d) If the insurance company charges \(\$ 3\) million for 84 months of insurance, how much profit does the company expect to make?

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-0.45 \leq z \leq 2.73) $$

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-2.18 \leq z \leq-0.42) $$

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