Question

A social skills training program was implemented with seven special needs students in a study to determine whether the program caused improvement in pre/post measures and behavior ratings. For one such test, the pre- and posttest scores for the seven students are given in the table. \(^{15}\) $$\begin{array}{lrr}\hline \text { Subject } & \text { Pretest } & \text { Posttest } \\\\\hline \text { Evan } & 101 & 113 \\\\\text { Riley } & 89 & 89 \\\\\text { Jamie } & 112 & 121 \\\\\text { Charlie } & 105 & 99 \\\\\text { Jordan } & 90 & 104 \\\\\text { Susie } & 91 & 94 \\\\\text { Lori } & 89 & 99 \\\\\hline\end{array}$$ a. What type of correlation, if any, do you expect to see between the pre- and posttest scores? Plot the data. Does the correlation appear to be positive or negative? b. Calculate the correlation coefficient \(r\). Is there a significant positive correlation?

Short answer

Answer: To determine if there is a significant positive correlation between the pretest and posttest scores, we need to calculate the correlation coefficient (r) and compare it to the critical value (approximately 0.786 for a sample size of 7 and a significance level of 0.05). If the calculated r is greater than the critical value, there is a significant positive correlation.

Step-by-step solution

To achieve this, we will plot the pretest scores (x-axis) against the posttest scores (y-axis) for each student. - Calculate the difference in scores for each student by subtracting the pretest score from the posttest score. - If the majority of the points have an increase in posttest scores, we would expect to see a positive correlation; if the majority have a decrease in posttest scores, we would expect a negative correlation. #b. Calculate the correlation coefficient r and determine if it is significant#

To find the correlation coefficient r, follow these steps: 1. Calculate the means of the pretest and posttest scores. 2. Calculate the standard deviation of the pretest and posttest scores. 3. Calculate the covariance using the formula: $$cov(X, Y) = \frac{\Sigma(x_i - \overline{x})(y_i - \overline{y})}{n}$$ 4. Calculate the correlation coefficient r using the formula: $$r = \frac{cov (X, Y)}{S_X S_Y}$$ 5. Compare the value of r to the critical value of a significant correlation using a table or calculator. For a sample size of 7 and a significance level of 0.05, the critical value is approximately 0.786. If the calculated r is greater than the critical value, conclude that there is a significant positive correlation.

Key concepts

Positive and Negative Correlation

Understanding the relationship between two variables is a cornerstone of statistical analysis. A correlation can either be positive, where an increase in one variable corresponds with an increase in the other, or negative, where an increase in one variable corresponds with a decrease in the other.

For example, if you were to study the relationship between hours studied and exam scores, you'd likely find that more hours studied is associated with higher exam scores, indicating a positive correlation. Conversely, consider a study on stress and health where higher stress levels might correlate with poorer health, revealing a negative correlation.

Standard Deviation

When you hear the term standard deviation in statistics, think of it as a measure of how spread out numbers are in a data set. The standard deviation tells us how much variation or 'dispersion' there is from the average (mean) value.

A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation points to values being spread out over a wider range. This concept is critical when calculating the correlation coefficient since it helps us understand how tightly the data points cluster around the line of best fit.

Covariance

Covariance is a measure that determines how much two random variables vary together. It's a key step in finding the correlation coefficient. If we're looking at two variables, let's say X and Y, their covariance would reflect whether an increase in X typically goes with an increase (or decrease) in Y.

The formula for covariance is:
\[ cov(X, Y) = \frac{\Sigma(x_i - \overline{x})(y_i - \overline{y})}{n} \]
where \( x_i \) and \( y_i \) are the individual sample points, and \( \overline{x} \) and \( \overline{y} \) are the respective means of X and Y. A positive value indicates a positive relationship, whereas a negative value denotes a negative relationship.

Significance of Correlation

Once we compute the correlation coefficient, it's not enough to just report its value. We must determine if the observed correlation is statistically significant, which means it's unlikely to have occurred by random chance.

The significance of a correlation coefficient is affirmed if it exceeds the critical value from the correlation coefficient table for a given alpha level (commonly set at 0.05 for a 95% confidence level). An alpha level represents the probability of rejecting the null hypothesis when it's actually true.

In simple terms, a significant correlation coefficient suggests a real association between variables, subject to the level of confidence we've set for the test.