For a random sample of 353 high school teachers, the...

Name: Problem 58, Chapter 11: Statistics for Business and Economics: Global Edition
Uploaded: 2022-09-29T15:31:21-08:00
Duration: 3 min 48 s
Channel: Rashmi Sinha
Description: For a random sample of 353 high school teachers, the correlation between annual raises and teaching evaluations was found to be 0.11 . Test the null hypothesis that these quantities are uncorrelated in the population against the alternative that the population correlation is positive.

For a random sample of 353 high school teachers, the correlation between annual raises and teaching evaluations was found to be 0.11 . Test the null hypothesis that these quantities are uncorrelated in the population against the alternative that the population correlation is positive.

Key Concepts

Test Statistic

In the context of testing a correlation coefficient, the test statistic is typically derived by transforming the sample correlation into a form that follows a known probability distribution, most commonly a t-distribution. This calculation accounts for the sample size and helps in assessing how extreme the observed correlation is under the null hypothesis.

Significance Level and p-value

The significance level is a threshold set by the researcher (commonly 0.05) to determine whether the observed results are statistically significant. The p-value is the probability of obtaining a test statistic as extreme as the observed value under the assumption that the null hypothesis is true. If the p-value is less than the significance level, it suggests that the observed correlation is unlikely to have happened by chance, leading to the rejection of the null hypothesis.

Degrees of Freedom

Degrees of freedom represent the number of independent values that can vary in the calculation of a statistic. For the test of a correlation coefficient, the degrees of freedom are determined by the sample size minus two, which adjusts the t-distribution used for the test statistic.

Alternative Hypothesis

The alternative hypothesis in a correlation test states that there is a non-zero relationship between the variables. When specified as positive, it means that the test is checking for evidence of a positive association between the two variables, where the true correlation coefficient is greater than zero.

Null Hypothesis

The null hypothesis in the context of correlation testing typically states that there is no linear relationship between the variables in the population. This means that the true population correlation coefficient is zero.

Correlation Coefficient

The correlation coefficient is a numerical summary of the linear relationship between two numerical variables. It ranges from -1 to 1, where a value close to 1 indicates a strong positive linear relationship, a value close to -1 indicates a strong negative linear relationship, and a value near 0 implies little or no linear association.

Hypothesis Testing

Hypothesis testing is a formal procedure used to evaluate whether the data provides sufficient evidence to reject a null hypothesis in favor of an alternative hypothesis. It involves computing a test statistic from the sample data and comparing it to a theoretical distribution to determine the likelihood of observing the data under the null hypothesis.

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