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The following table lists the midterm and final exam scores for seven students in a statistics class. $$ \begin{array}{l|lllllll} \hline \text { Midterm score } & 79 & 95 & 81 & 66 & 87 & 94 & 59 \\ \hline \text { Final exam score } & 85 & 97 & 78 & 76 & 94 & 84 & 67 \\ \hline \end{array} $$ a. Do you expect the midterm and final exam scores to be positively or negatively related? b. Plot a scatter diagram. By looking at the scatter diagram, do you expect the correlation coefficient between these two variables to be close to zero, 1 , or \(-1\) ? c. Find the correlation coefficient. Is the value of \(r\) consistent with what you expected in parts a and \(\mathrm{b}\) ? d. Using the \(1 \%\) significance level, test whether the linear correlation coefficient is positive.

Step by step solution

01

Interpret Data

To understand if the midterm and final scores are positively or negatively related, look at the data provided. If a student who scored high on the midterm also scored high on the final, and a student who scored low on the midterm also scored low on the final, it indicates a positive relationship. If the opposite is true, it indicates a negative relationship. From observing the given data, it seems there is a positive relationship.

02

Plot a Scatter Diagram

A scatter diagram can be plotted using any statistical software or graphing calculator. On the horizontal axis, plot the midterm scores, and on the vertical axis, plot the final exam scores. The points should have a somewhat linear upward trend if there is a positive relationship. It's expected that the correlation coefficient is closer to 1 due to this observed trend.

03

Calculate Correlation Coefficient

Correlation coefficient \(r\) is given by the formula \[r = \frac{\sum_xy - n \bar{x}\bar{y}}{\sqrt{(\sum_x^2 - n \bar{x}^2)(\sum_y^2 - n \bar{y}^2)}}\] where \(x\) represents the midterm scores and \(y\) represents the final exam scores, \(n\) is the number of pairs (in this case, 7), and \(\bar{x}\) and \(\bar{y}\) are the respective means. Calculate \(\sum_xy\), \(\sum_x^2\), \(\sum_y^2\), \(\bar{x}\), and \(\bar{y}\), and then substitute these values into the formula to find \(r\).

04

Interpret Correlation Coefficient

The value obtained for the correlation coefficient, \(r\), will be compared with your expectations from Steps 1 and 2. If it's closer to 1 and thus consistent with your expectations, it means the correlation is indeed positive.

05

Test of Significance

The significance of the correlation can be tested using the correlation coefficient, the number of pairs and the significance level. The critical value for a positive correlation with \(n-2\) degrees of freedom at the 1% significance level can be found in tables of the t-distribution, or calculated using statistical software. If the calculated \(r\) exceeds the critical value, it shows the correlation is positive at the 1% level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is an important visual tool used to determine the type of relationship between two variables. In this case, it helps us to see how midterm scores relate to final exam scores for a group of students. To create a scatter plot:

• Place the midterm scores on the horizontal axis (x-axis).
• Place the final exam scores on the vertical axis (y-axis).
• Plot a point for each student based on their scores, resulting in seven points total.

If there's a general upward trend in the plotted points, this suggests a positive relationship, meaning as midterm scores increase, final scores tend to increase as well. This visual representation gives a preliminary idea of whether there's a correlation between the two sets of scores.

Linear Relationship

A linear relationship between two variables suggests that they change at a consistent rate with respect to each other. In simpler terms, if midterm scores increase by a certain amount, we expect final exam scores to increase or decrease by a predictable amount as well.

• In a positive linear relationship, both variables increase together.
• In a negative linear relationship, one variable increases while the other decreases.
• If the variables do not show such predictable patterns, the relationship may not be linear.

In the exercise, a positive linear relationship is expected as students who perform well on the midterm tend to also do well on the final exam, as observed from the data.

Correlation Coefficient

The correlation coefficient, denoted as \( r \), quantifies the strength and direction of a linear relationship between two variables. It can range from -1 to 1:

• \( r = 1 \): Perfect positive linear relationship.
• \( r = -1 \): Perfect negative linear relationship.
• \( r = 0 \): No linear relationship.

To calculate \( r \), use the formula:\[ r = \frac{\sum xy - n \bar{x}\bar{y}}{\sqrt{(\sum x^2 - n \bar{x}^2)(\sum y^2 - n \bar{y}^2)}} \]where \( x \) and \( y \) represent the scores, and \( n \) is the number of observations. By substituting the values from the exercise, the correlation coefficient will confirm the expected relationship indicated by the scatter plot.

Significance Testing

Significance testing helps determine if the observed correlation is statistically meaningful or if it could have happened by chance. In this context, we use a hypothesis test at a 1% significance level to check if the positive correlation is valid.

• First, establish the null hypothesis: there is no linear relationship.
• Calculate the test statistic using the correlation coefficient.
• Compare this to the critical value from a t-distribution table adjusted for 5 degrees of freedom (as there are 7 observations).

If the test statistic exceeds the critical value, we can reject the null hypothesis, affirming that a positive linear relationship likely exists between the midterm and final scores at a 1% significance level.