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. An environmentalist wants to determine the relationships between the numbers (in thousands) of forest fires over the year and the number (in hundred thousands) of acres burned. The data for 8 recent years are shown. Describe the relationship. $$ \begin{array}{l|llllllll} \text { Number of fires } x & 72 & 69 & 58 & 47 & 84 & 62 & 57 & 45 \\ \hline \text { Number of } & & & & & & & & \\ \text { acres burned } y & 62 & 42 & 19 & 26 & 51 & 15 & 30 & 15 \end{array} $$

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.
An environmentalist wants to determine the relationships between the numbers (in thousands) of forest fires over the year and the number (in hundred thousands) of acres burned. The data for 8 recent years are shown. Describe the relationship.
$$
\begin{array}{l|llllllll}
\text { Number of fires } x & 72 & 69 & 58 & 47 & 84 & 62 & 57 & 45 \\
\hline \text { Number of } & & & & & & & & \\
\text { acres burned } y & 62 & 42 & 19 & 26 & 51 & 15 & 30 & 15
\end{array}
$$

Key Concepts

Test for Significance of Correlation

Testing the significance of a correlation involves calculating a test statistic (often using a t-test for correlation) based on the sample size and the correlation coefficient, and then comparing it with a critical value from a statistical table at the chosen significance level. This process helps determine if the observed correlation is statistically meaningful.

Relationship Interpretation

Interpreting the relationship involves summarizing the nature of the association between the variables after data analysis. This includes noting if the relationship is positive or negative, linear or non-linear, and whether the strength of the correlation is weak, moderate, or strong, thus providing context for the practical implications of the statistical findings.

Significance Level

The significance level, denoted as alpha (?), is the threshold used to determine when to reject the null hypothesis. Commonly set at 0.05, it represents a 5% risk of committing a Type I error—rejecting the null hypothesis when it is actually true.

Hypothesis Testing

Hypothesis testing is a systematic process for making inferences about a population based on sample data. In the context of correlation, it involves comparing the calculated test statistic against critical values to decide whether the observed relationship is statistically significant or could have occurred by chance.

Null and Alternative Hypotheses

When testing the significance of a correlation coefficient, the null hypothesis typically states that there is no linear relationship (correlation coefficient equals zero), while the alternative hypothesis suggests that there is a non-zero correlation. This framework forms the basis for statistical inference in assessing relationships between variables.

Correlation Coefficient

The correlation coefficient is a statistic that quantifies the direction and strength of a linear relationship between two quantitative variables. Values range from -1 to 1, where values close to 1 or -1 indicate strong positive or negative linear relationships respectively, and values near 0 suggest minimal or no linear association.

Scatter Plot

A scatter plot is a visual tool that displays pairs of numerical data, typically with one variable on the horizontal axis and the other on the vertical axis. It is used to identify patterns, trends, clusters, or potential outliers in the data and provides an intuitive way to assess the possible relationship between the two variables.

Transcript

00:01 Here's a solution for the number of fires and number of acres burned.

00:03 The first part we're going to just do a quick scatter plot.

00:06 So if we go to stat and then edit, you can type in your data values there, okay? and then go to second y equals just to make sure that stat plots on.

00:17 It's a scatter plot.

00:18 And then you can graph it, and chances are you're not going to see it, which is fine.

00:20 Just go to zoom 9, and then you can see it looks to be fairly linear.

00:25 But let's double check and make sure.

00:26 So let's go to calc and sat calc and then lin reg.

00:29 So we want to see if this is a linear relationship.

00:32 So let's go to calculate and you get an r, that's the important statistic there, 0 .7705, and an r squared of 0 .59.

00:41 So let's write those down.

00:43 So r equals 0 .7505.

00:46 The null hypothesis is that row equals 0, and then the alternative is that row is not equal to 0.

00:52 So that means that the null is that there is no significant linear relationship, and the alternative is that there is a significant linear relationship.

01:02 So then your statistic, if we take the r, the .7705, and then we divide by the square root of 1 minus r squared, which is 0 .5937, and then we divide that by n minus 2 degrees of freedom...

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