Question

True/false: The sampling distribution of \(\mathrm{r}=.8\) becomes normal as \(\mathrm{N}\) increases.

Short answer

True. The sampling distribution becomes normal as N increases due to the CLT.

Step-by-step solution

Understand Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic obtained from a random sample. In this case, it's the distribution of the correlation coefficient, \(r\).

Recall the Central Limit Theorem

The Central Limit Theorem (CLT) states that, given a sufficient sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the original distribution.

Apply the CLT to Correlation

The CLT not only applies to means but also to sample proportions and correlation coefficients. As the sample size (N) increases, the distribution of the sample correlation coefficient \(r\) tends to become normal.

Consider the Given Correlation

Given \(r = 0.8\), a high positive correlation, as sample size increases, the distribution of \(r\) will approximate a normal distribution due to the application of the CLT.

Conclusion

Because the CLT holds for correlation coefficients, the statement that the sampling distribution of \(r = 0.8\) becomes normal as \(N\) increases is true.

Key concepts

Understanding Sampling Distribution

Sampling distribution is a concept that illustrates how the distribution of a specific statistic, such as a sample mean or a correlation coefficient, behaves when you repeatedly sample from a population. In more simple terms, if we were to take a lot of different samples from the same population and calculate the correlation coefficient for each, we would end up with a distribution of those correlation coefficients.

• It helps us to understand the variation and reliability of a sample statistic.
• It forms the basis for making inferences about the overall population.
• The shape of this distribution can provide insights into the nature of the sample statistic.

Overall, the sampling distribution provides crucial insights and forms a bridge between the sample data we have and the population we are interested in. Understanding this concept is key to applying statistical principles correctly.

What is a Correlation Coefficient?

The correlation coefficient is a numerical measure that expresses the degree of a linear relationship between two variables. It is usually represented by the symbol \(r\) and lies between -1 and 1. Here's what different values of \(r\) indicate:

• \(r = 1\): Perfect positive correlation; both variables move in the same direction.
• \(r = -1\): Perfect negative correlation; one variable increases while the other decreases.
• \(r = 0\): No correlation; the two variables do not have a linear relationship.

In the context of sampling distributions, the term refers to the fact that correlation coefficients can take on a range of values, and how these values are distributed across multiple samples from the population. Calculating the correlation coefficient allows us to quantify the strength and direction of a relationship between two continuous variables.

The Importance of Sample Size

Sample size, denoted as \(N\), is a crucial element in statistics. It represents the number of observations or data points in a sample. The role of sample size is significant when considering the reliability and validity of statistical conclusions.

• Larger sample sizes generally provide more accurate estimates of a population parameter.
• They decrease the margin of error and increase the confidence level of the results.
• In the context of the Central Limit Theorem, a larger \(N\) means the sampling distribution will more closely resemble a normal distribution.

The Central Limit Theorem's behavior emphasizes the importance of a sufficiently large \(N\), especially when dealing with distributions that aren't initially normal. This is why sample size is such a pivotal part of designing experiments and analyzing data.

Understanding Probability Distribution

A probability distribution is a mathematical function that describes the likelihood of different outcomes in a random experiment. It's the foundation on which probability theory is built. There are various types of probability distributions depending on the nature of data, such as discrete or continuous.

• Discrete distributions (like the binomial distribution) involve outcomes that are discrete, like rolling a die.
• Continuous distributions (like the normal distribution) involve outcomes that are continuous, like the height of people.
• Probability distributions can be used to calculate probabilities, expected values, and other relevant statistical parameters, helping to predict how often an event is likely to occur.

In the realm of sampling distributions, probability distribution becomes essential, as it helps in modeling the variability of sample statistics. This modeling, especially as it relates to the Central Limit Theorem, is fundamental to making inferences or predictions about a broader population.