4.	Suppose that the health psychologist repeated the same study by collecting new data from another group of people. This time, she found r = −0.65 and ΣZXZY = −8.45. How many people did she have in this new sample? current data set: x data set: 9, 8, 9, 5, 8, 9, 7, 5, 7, 8, y data set: 4, 3, 3, 1, 3, 3, 2, 4, 3, 3

To determine the number of people (\(n\)) in this new sample, we can use the formula that connects the correlation coefficient (\(r\)) with the sum of the products of standardized \(Z\)-scores (\(\Sigma Z_X Z_Y\)):

\[
r = \frac{\Sigma Z_X Z_Y}{n - 1}
\]

Given the following:

- \(r = -0.65\)
- \(\Sigma Z_X Z_Y = -8.45\)

We substitute the known values into the formula and solve for \(n\):

\[
-0.65 = \frac{-8.45}{n - 1}
\]

Now, solve for \(n - 1\):

\[
n - 1 = \frac{-8.45}{-0.65} = \frac{8.45}{0.65}
\]

Calculate the right-hand side:

\[
n - 1 = 13
\]

Finally, solve for \(n\):

\[
n = 13 + 1 = 14
\]

So, the psychologist had 14 people in the new sample.

Would you like more details on this process, or do you have any other questions?

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Here are five follow-up questions related to correlation and sample size:

1. How does the value of \(r\) affect the interpretation of a correlation study?
2. What is the significance of a negative correlation coefficient like \(r = -0.65\)?
3. How do the \(Z\)-scores influence the correlation coefficient?
4. What would happen to the correlation if more people were added to the sample?
5. How is the strength of the relationship between variables affected by the sample size?

**Tip:** When interpreting correlation, remember that values closer to -1 or 1 indicate stronger relationships, while values closer to 0 indicate weaker relationships.

Suppose that the health psychologist repeated the same study by collecting new data from another group of people. This time, she found r = −0.65 and ΣZXZY = −8.45. How many people did she have in this new sample? Current data set: x data set: 9, 8, 9, 5, 8, 9, 7, 5, 7, 8, y data set: 4, 3, 3, 1, 3, 3, 2, 4, 3, 3.

Find out how to calculate the sample size using the correlation coefficient (r = -0.65) and the sum of Z-scores (ΣZXZY = -8.45). This problem uses statistical concepts, including the relationship between Z-scores and correlation. Learn how to solve for n with a step-by-step explanation, suitable for high school students.

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