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. School administrators wondered whether class size and grade achievement (in percent) were related. A random sample of classes revealed the following data. Are the variables linearly related? $$ \begin{array}{l|cccccc} \text { No. of students } & 15 & 10 & 8 & 20 & 18 & 6 \\ \hline \text { Avg. grade }(\%) & 85 & 90 & 82 & 80 & 84 & 92 \end{array} $$

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.
School administrators wondered whether class size and grade achievement (in percent) were related. A random sample of classes revealed the following data. Are the variables linearly related?
$$
\begin{array}{l|cccccc}
\text { No. of students } & 15 & 10 & 8 & 20 & 18 & 6 \\
\hline \text { Avg. grade }(\%) & 85 & 90 & 82 & 80 & 84 & 92
\end{array}
$$

Key Concepts

Scatter Plot

A scatter plot is a graphical representation of the relationship between two quantitative variables. It uses dots to represent the data points, allowing one to visually assess the possible association or pattern between the variables, such as a trend, clustering, or outliers.

Correlation Coefficient

The correlation coefficient is a numerical measure that quantifies the strength and direction of the linear relationship between two quantitative variables. Its value ranges from -1 to 1, where values close to 1 indicate a strong positive linear relationship, values close to -1 indicate a strong negative linear relationship, and values around 0 suggest little to no linear association.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. In the context of correlation, this involves formulating a null hypothesis, which typically states that there is no linear association between the variables (i.e., the correlation coefficient is zero), against an alternative hypothesis that there is a linear relationship.

Significance Level

The significance level, often denoted by ? (alpha), is the threshold used to decide whether a statistical result is significant. In hypothesis testing, if the p-value is less than the significance level, the result is considered statistically significant, leading to rejection of the null hypothesis.

Linear Relationship

A linear relationship between two variables means that as one variable changes, the other variable tends to change in a constant proportion. This can be identified both visually through a scatter plot and quantitatively through measures such as the correlation coefficient. It implies that the association between the variables can be described with a straight line.

Transcript

00:01 Here's the solution to the number of students in the average grade example.

00:03 We're supposed to draw a scatter plot and then compute the r, correlation coefficient, and then run a hypothesis test.

00:10 Okay, so for the scatter plot, i'm going to use, actually i'm going to use graph and calculator on all this.

00:15 So if you go to stat and then edit, you can pack in your data values, and then go ahead and graph it, and you'll need to make sure your stat plots on and it's on that scatter plot.

00:26 And if you can't see it, that's fine.

00:27 If you go to zoom 9, you should be able to see it.

00:29 So it looks decent except for this one little point out here.

00:31 That seems to be a bit off.

00:34 So that's the scatter plot.

00:37 So we're done with that.

00:38 Now let's find the r.

00:39 So if you go to stat and then calculate, and we're going to go to option for the lin reg ax plus b, because we're going to see if this is linear regression.

00:48 And we're going to calculate, and then there's your r.

00:50 So negative .673.

00:53 Okay, so that's part b.

00:55 So r equals negative 0 .677.

01:01 Your null hypothesis is that row equals 0, and the alternative is row is not equal to 0...

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