Compute the value of the Pearson Point Bi-serial correlation coefficient from the given set of information: 	Yˉ0=937.5, Yˉ1=455, sX=217,ρ0=0.2, ρ1=0.8

To compute the Pearson Point Bi-serial correlation coefficient (\( r_{pb} \)), we will use the following formula:

\[ r_{pb} = \frac{Y_1 - Y_0}{s_Y} \cdot \sqrt{\frac{p_0 \cdot p_1}{p_0 + p_1}} \]

where:
- \( Y_0 \) and \( Y_1 \) are the mean values of \( Y \) for the binary variable \( X \) categories 0 and 1, respectively.
- \( s_Y \) is the standard deviation of \( Y \).
- \( p_0 \) and \( p_1 \) are the proportions of observations in categories 0 and 1, respectively.

Given:
- \( Y_0 = 937.5 \)
- \( Y_1 = 455 \)
- \( s_Y = 217 \)
- \( p_0 = 0.2 \)
- \( p_1 = 0.8 \)

Plugging in the values:

\[ r_{pb} = \frac{455 - 937.5}{217} \cdot \sqrt{\frac{0.2 \cdot 0.8}{0.2 + 0.8}} \]

First, compute the difference in means:

\[ 455 - 937.5 = -482.5 \]

Next, compute the term under the square root:

\[ \sqrt{\frac{0.2 \cdot 0.8}{1}} = \sqrt{0.16} = 0.4 \]

Finally, compute \( r_{pb} \):

\[ r_{pb} = \frac{-482.5}{217} \cdot 0.4 = -2.2244 \cdot 0.4 = -0.88976 \]

Therefore, the Pearson Point Bi-serial correlation coefficient is approximately \( -0.88976 \).

Do you want more details or have any questions about this computation? 

Here are 5 more questions you might ask:
1. How is the standard deviation calculated for a dataset?
2. What is the difference between Pearson correlation and point-biserial correlation?
3. Can you explain the assumptions behind the Pearson correlation coefficient?
4. How do you interpret the value of a correlation coefficient?
5. What are some limitations of using correlation coefficients in data analysis?

**Tip:** Always check if your data meets the assumptions required for the statistical methods you are using to ensure valid results.

Learn how to compute the Pearson Point Bi-serial correlation coefficient using mean values, standard deviation, and category proportions. This problem involves calculating statistical correlation coefficients based on given data.

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