Perform the following steps. a. Draw the scatter plot for...

Perform the following steps. a. Draw the scatter plot for the variables. b. Compute the value of the correlation coefficient. c. State the hypotheses. d. Test the significance of the correlation coefficient at $\alpha=0.05,$ using Table I. e. Give a brief explanation of the type of relationship. Assume all assumptions have been met. The director of an alumni association for a small college wants to determine whether there is any type of relationship between the amount of an alumnus's contribution (in dollars) and the number of years the alumnus has been out of school. The data follow. $$ \begin{array}{l|cccccc} \text { Years } x & 1 & 5 & 3 & 10 & 7 & 6 \\ \hline \text { Contribution } y & 500 & 100 & 300 & 50 & 75 & 80 \end{array} $$

Perform the following steps.
a. Draw the scatter plot for the variables.
b. Compute the value of the correlation coefficient.
c. State the hypotheses.
d. Test the significance of the correlation coefficient at $\alpha=0.05,$ using Table I.
e. Give a brief explanation of the type of relationship. Assume all assumptions have been met.
The director of an alumni association for a small college wants to determine whether there is any type of relationship between the amount of an alumnus's contribution (in dollars) and the number of years the alumnus has been out of school. The data follow.
$$
\begin{array}{l|cccccc}
\text { Years } x & 1 & 5 & 3 & 10 & 7 & 6 \\
\hline \text { Contribution } y & 500 & 100 & 300 & 50 & 75 & 80
\end{array}
$$

Key Concepts

Scatter Plot

A scatter plot is a graphical tool used to display the relationship between two quantitative variables by placing points on a coordinate plane. This visualization helps in detecting patterns, trends, and potential outliers that might influence further statistical analysis.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables. Its value ranges from -1 to 1, indicating strong negative to strong positive linear relationships, with 0 signifying no linear correlation.

Hypothesis Testing for Correlation

When testing for correlation, hypotheses are established to determine if the observed linear relationship is statistically significant. Typically, the null hypothesis states that there is no correlation (i.e., a correlation coefficient of 0), while the alternative hypothesis suggests that a non-zero correlation exists.

Significance Level

The significance level, denoted by alpha (?), defines the probability threshold for rejecting the null hypothesis. It represents the acceptable risk of committing a Type I error, and common levels such as 0.05 are used to assess whether the observed relationship is unlikely to have occurred by chance.

Testing the Significance of the Correlation

This key concept involves using a statistical test, such as a t-test for correlation, which calculates a test statistic that is compared against critical values from a t-distribution at the chosen significance level. This process determines if the observed correlation is statistically significant and not due to random variations in the data.

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