Question

What is the shape of the sampling distribution of \(r\) ? In what way does the shape depend on the size of the population correlation?

Short answer

The shape of the sampling distribution of \( r \) is influenced by population correlation \( \rho \) and sample size, becoming more normal with larger samples.

Step-by-step solution

Understanding the Sampling Distribution

The sampling distribution of the sample correlation coefficient, denoted as \( r \), is the probability distribution of the \( r \) values obtained from all possible samples of a specific size from a population.

Shape of the Distribution

The shape of the sampling distribution of \( r \) depends on the population correlation coefficient, denoted as \( \rho \). When \( \rho = 0 \), the distribution tends to be nearly normal, especially as the sample size increases. However, when \( \rho e 0 \), the distribution becomes skewed.

Impact of Sample Size

As the sample size increases, the distribution of \( r \) becomes more normal regardless of the actual population correlation \( \rho \). Larger samples lead to a narrower distribution with reduced variability.

Effect of Population Correlation

For larger values of \( \rho \) (close to -1 or 1), the distribution of \( r \) exhibits more skewness. The skewness decreases with an increase in the sample size, leading the distribution closer to a normal distribution.

Key concepts

Sample Correlation Coefficient

The sample correlation coefficient, denoted as \( r \), is a numerical measure that assesses the strength and direction of the linear relationship between two variables in a sample. This concept is fundamental in statistics and is often used in studies where researchers select samples from larger populations to make inferences.

Interpretation of \( r \)
- Values close to 1 indicate a strong positive linear relationship.
- Values close to -1 indicate a strong negative linear relationship.
- Values around 0 suggest a weak or no linear relationship.

The sample correlation coefficient is vital as it provides insights into the data patterns, helping researchers understand how variables are related in a sample.

Population Correlation Coefficient

The population correlation coefficient, represented by \( \rho \) (Greek letter rho), reflects the true correlation of variables in the whole population. Unlike the sample correlation coefficient \( r \), which is an estimate, \( \rho \) is the actual measure of linear association between variables across the entire data set.

Importance of \( \rho \)
- It serves as a benchmark to evaluate the sample correlation coefficient.
- Understanding \( \rho \) helps in predicting and making decisions about the population.

As researchers often deal with samples instead of whole populations, the difference in values between \( r \) and \( \rho \) can provide insight into the reliability of the sample data.

Normal Distribution

Normal distribution is a continuous probability distribution that is symmetric around the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. It is essential in statistics due to its unique properties.

Key Characteristics
- The mean, median, and mode of a normal distribution are all equal.
- It has a bell-shaped curve, where most of the data points cluster around the center.
- The distribution is defined by two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)).

Understanding normal distribution is crucial as it helps describe how the sample correlation coefficient \( r \) behaves under different population correlation coefficient \( \rho \) scenarios. When \( \rho =0 \), and the sample size is large, \( r \)'s distribution becomes approximately normal, providing a useful approximation for hypothesis testing and inference.

Skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It can indicate the direction and degree of outlier-prone observations.

Types of Skewness
- **Positive Skewness:** When the tail on the right side of the distribution is longer or fatter, indicating data are skewed to the right.
- **Negative Skewness:** When the tail on the left side is longer or fatter, indicating data are skewed to the left.

Skewness plays a role in interpreting the shape of the sampling distribution of \( r \). When \( \rho eq 0 \), the distribution of \( r \) becomes skewed. The degree of skewness decreases with an increase in sample size, which gradually makes the distribution of \( r \) more normal.