In the study of 49 countries discussed in Example 11.4, the...

Name: Problem 57, Chapter 11: Statistics for Business and Economics: Global Edition
Uploaded: 2022-11-09T22:31:35-08:00
Channel: Victor Salazar
Description: In the study of 49 countries discussed in Example 11.4, the sample correlation between the experts' political riskiness score and the infant mortality rate in these countries was 0.75 . Test the null hypothesis of no correlation between these quantities against the alternative of positive correlation.

In the study of 49 countries discussed in Example 11.4, the sample correlation between the experts' political riskiness score and the infant mortality rate in these countries was 0.75 . Test the null hypothesis of no correlation between these quantities against the alternative of positive correlation.

Key Concepts

Degrees of Freedom

Degrees of freedom in the context of correlation testing refer to the number of independent pieces of information available to estimate the parameter. For the t-test of a correlation coefficient, the degrees of freedom are calculated as the sample size minus 2, reflecting the estimation of two means (one for each variable).

t-Test for Correlation

When testing the significance of a correlation coefficient, the test statistic can be computed using a t-transformation, which is given by t = r * sqrt((n - 2) / (1 - r²)), where n is the sample size. This t-statistic follows a t-distribution with n - 2 degrees of freedom, and it allows for determining the probability of observing the given correlation under the null hypothesis.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, with values near 0 indicating little or no linear relationship, values near 1 indicating a strong positive relationship, and values near -1 indicating a strong negative relationship.

Hypothesis Testing

Hypothesis testing is a formal statistical procedure used to decide whether the data at hand sufficiently support a particular hypothesis about a population parameter. In the context of correlation, it involves testing if the observed correlation could have arisen by chance under the assumption that the true correlation is zero.

Null and Alternative Hypotheses

The null hypothesis (H0) typically posits that there is no effect or no relationship—in this case, that the population correlation coefficient is zero. The alternative hypothesis (Ha) is what the researcher wants to support, such as a positive association between the variables.

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