Question

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if there is evidence that a correlation between height and salary is significant (that is, different than zero ).

Short answer

The relevant parameters are height and salary. The null hypothesis (H0) is that there is no correlation between height and salary (\( \rho = 0 \)). The alternative hypothesis (H1) is that the correlation between height and salary is significantly different than zero (\( \rho \neq 0 \)).

Step-by-step solution

Definition of Parameters

The parameters in this case study are height and salary. They both constitute a correlation relationship.

Setting up the Null Hypothesis

The null hypothesis (H0) is often a statement of no effect or no relationship. So in this case, the null hypothesis would be: There is no correlation between height and salary, which translates to a correlation coefficient of 0.

Setting up the Alternative Hypothesis

The alternative hypothesis (H1) is the statement that directly contradicts the null hypothesis. What we want to prove is that a correlation between height and salary exists (meaning the correlation is significantly different than zero). Hence, the alternative hypothesis would be that there is a non-zero correlation.

Key concepts

Null Hypothesis

In any statistical analysis, the null hypothesis is critical as it is the claim being tested. The null hypothesis asserts that there is no association or effect between the variables of interest. In the context of the correlation between height and salary, the null hypothesis (\( H_0 \)) posits that the correlation coefficient, which measures the strength and direction of this relationship, is equal to zero. This means there is no evidence to suggest any relationship between height and salary in the population.

It's like a base assumption that we start with, and our objective in hypothesis testing is to challenge this assumption. It places the burden of proof on those who claim that a relationship exists, requiring them to demonstrate this with sufficient evidence from data. Setting up a clear null hypothesis is crucial for a strong foundation in any statistical test.

Alternative Hypothesis

The alternative hypothesis (\( H_1 \) or (\( H_a \)) serves as the counter-claim to the null hypothesis. It's what you suspect might be true and are seeking evidence to support. In the scenario where we're investigating a relationship between height and salary, the alternative hypothesis posits that there exists a non-zero correlation: essentially, height does have some effect on salary.

This hypothesis doesn't necessarily specify whether the effect is positive or negative, only that there's some statistically significant relationship to be observed. The alternative hypothesis fosters the investigative nature of statistical testing, where we're set to either reject the null hypothesis in favor of the alternative or fail to find enough evidence to do so.

Correlation Coefficient

The correlation coefficient is a statistical measure that calculates the strength of the relationship between two variables. It ranges from -1 to 1, where 1 represents a perfect positive correlation, 0 represents no correlation, and -1 represents a perfect negative correlation.

This coefficient is the crux of our statistical test in the given exercise. If it significantly deviates from zero (in either a positive or negative direction), it suggests that as one variable increases, the other variable also tends to increase (or decrease, if the correlation is negative). It's this value that is tested against the null hypothesis when we assess the significance of the relationship between height and salary.

Parameters in Statistics

Parameters in statistics refer to the defining characteristics or measurable factors of a population that help in concluding from sampled data. For example, the mean of a population is a parameter that summarizes the average value of that population.

In the exercise at hand, height and salary are the key parameters as they define the aspects of the population we're examining for a correlation. Identifying the correct parameters is essential in setting up the hypotheses and conducting an accurate test. In practice, we often estimate parameters through a statistic, a measure computed from sample data, since having data for an entire population is rarely possible.