Question

When a researcher selects all possible pairs of samples from a population in order to find the difference between the means of each pair, what will be the shape of the distribution of the differences when the original distributions are normally distributed? What will be the mean of the distribution? What will be the standard deviation of the distribution?

Short answer

The distribution is normal with mean 0 and standard deviation \( \sigma \sqrt{\frac{2}{n}} \).

Step-by-step solution

Understanding the Task

We need to determine the shape, mean, and standard deviation of the distribution of differences between means of pairs of samples, given that the original population distribution is normal.

Shape of the Distribution

When samples are taken from a normally distributed population, the differences between the means of these samples will also be normally distributed. This is because the normal distribution characteristic is preserved in the difference of normal random variables.

Mean of the Distribution

For any pair of samples taken from a population, the expected (mean) difference between the means of these samples is zero. This is because each sample mean is an unbiased estimator of the population mean.

Standard Deviation of the Distribution

To find the standard deviation of the distribution of differences, if each sample has size n from a population with standard deviation , we use the formula for the standard deviation of the difference between two samples: \( \sigma_{\bar{X}_1 - \bar{X}_2} = \sigma \sqrt{\frac{2}{n}} \). This formula arises from the standard deviation of the normal distribution and properties of variance.

Key concepts

Sampling Distribution

Sampling distribution refers to the probability distribution of a statistic obtained from many samples drawn from a specific population. When we talk about a sampling distribution of the mean, we are referring to the distribution of sample means gathered from multiple sample groups within the same population. This concept helps statisticians understand how the sample mean behaves compared to the population mean.

• This distribution provides valuable insights because it allows calculations about the probability of obtaining certain sample results.
• Understanding the shape of the sampling distribution helps in making predictions and inferences about the population.

In the given exercise, the researcher is interested in the distribution of differences between means for pairs of samples. When the original population is normally distributed, this sampling distribution also tends to display a normal shape.
This characteristic stems from the consistency of normal distribution properties when dealing with means.

Mean Difference

The mean difference is an important concept in statistics, especially when comparing two groups. It represents the average of all differences between the pairs of sample means. In the exercise, one must calculate the mean difference when the samples are drawn from a normally distributed population. Due to the properties of normal distribution and unbiased estimators:

• The mean difference is expected to be zero.
• Each sample mean serves as a reliable representation of the population mean.

Since any two samples are being compared, the law of large numbers suggests that the average outcome of repeated experiments gets closer to the expected value.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. When applied to a distribution's mean differences, it tells us how scattered these differences are. For the setup discussed in the exercise, the standard deviation of the distribution of differences between two sample means is calculated with the formula:\[\sigma_{\bar{X}_1 - \bar{X}_2} = \sigma \sqrt{\frac{2}{n}} \]where \( \sigma \) is the standard deviation of the population, and \( n \) is the size of each sample.

• This reveals how spread out the differences are, despite the expectation of a mean difference of zero.
• The larger the sample size, \( n \), the less variability there is among the mean differences.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics asserting that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, no matter the shape of the original population distribution.
In simpler terms, regardless of whether the population distribution is normal, skewed, or of any other shape, the distribution of the sample means tends to be normal. However, our exercise begins with a normally distributed population. Thus, when you draw pairs of samples and calculate their mean differences:

• The distribution of those differences shares the normal shape due to the inheritance of properties from the original population.
• This makes it easier to perform statistical tests and draw conclusions.

The CLT aids in the reliability of sample-based inferences about the population, highlighting the power of sample size in achieving a normal distribution.