Question

If numerous samples of \(\mathrm{N}=15\) are taken from a uniform distribution and a relative frequency distribution of the means is drawn, what would be the shape of the frequency distribution?

Short answer

The frequency distribution will be approximately normal.

Step-by-step solution

Understand the Central Limit Theorem

The Central Limit Theorem (CLT) states that if you take samples of a sufficiently large size from a population with any shape distribution, the sampling distribution of the sample means will be approximately normally distributed. This applies regardless of the original distribution shape as long as the sample size is sufficiently large.

Determine Sample Size Adequacy

The sample size here is 15. While it's generally accepted that a sample size of 30 or more is sufficient for the CLT to hold, in many practical applications and tests, a sample size of 15 is often large enough to approximate normality, especially if the underlying distribution is not too extreme.

Apply the Central Limit Theorem

Given the problem, the original samples are taken from a uniform distribution. The uniform distribution is symmetric and bounded, which makes it less extreme, allowing the CLT to hold even with smaller sample sizes like 15. Thus, the distribution of the sample means would tend to be normal.

Conclusion about the Shape

The relative frequency distribution of the means from samples of size 15 taken from a uniform distribution will be approximately normally distributed due to the Central Limit Theorem.

Key concepts

Sampling Distribution

A sampling distribution refers to the probability distribution of a given statistic based on a random sample. It helps us understand the behavior of sample statistics as they relate to the corresponding population parameters. When we talk about sample means in particular, we are interested in how these means vary from one sample to another across numerous samples.

• The Central Limit Theorem is a key concept that informs us about the shape and properties of the sampling distribution.
• If we draw samples from any population distribution, the sampling distribution of the means will approximate a normal distribution as the sample size becomes larger.
• This approximation holds true irrespective of the shape of the original population distribution, even if it's non-normal.
• Sampling distributions play a crucial role in statistical inference, enabling estimations of population parameters through sample data.

Thus, understanding sampling distributions is fundamental when interpreting sample data and making predictions about a population.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetrical bell-shaped curve. It is defined by its mean and standard deviation, with most data points clustering around the mean.

• Due to its properties, the normal distribution is often referred to as a 'Gaussian distribution'.
• It serves as an important model for many natural phenomena in various fields, from biology to finance.
• One remarkable property is that the normal distribution is completely specified by its mean (\( \mu \)) and standard deviation (\( \sigma \)).
• As the Central Limit Theorem states, the means of samples of a sufficiently large size will form a normal distribution, regardless of the population's original distribution.

This makes the normal distribution a powerful tool in inferential statistics, allowing for the testing of hypotheses and aiding in the calculation of confidence intervals.

Uniform Distribution

Uniform distribution is a type of probability distribution in which all outcomes are equally likely. It is often visualized as a rectangle in a probability density function, where the height represents equal likelihood across a defined range.

• Unlike the normal distribution, the uniform distribution does not have the bell curve shape as it remains flat and constant.
• This distribution is described by two parameters: a lower bound (\( a \)) and an upper bound (\( b \)), with mean (\( \mu = \frac{a+b}{2} \)) and variance (\( \sigma^2 = \frac{(b-a)^2}{12} \)).
• Despite its simplicity, the uniform distribution can serve as a good model for phenomena where each possible outcome is equally probable.
• As seen in the Central Limit Theorem, even when the original distribution of sample data is uniform, the distribution of sample means will tend toward a normal distribution as the sample size increases.

This flexible property allows us to effectively utilize uniform distributions in statistical sampling and simulation scenarios.