Compute the value of the 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 Point Bi-serial correlation coefficient, we use the following formula:

\[ r_{pb} = \frac{Ȳ_1 - Ȳ_0}{s_Y} \sqrt{\frac{ρ_0ρ_1}{n}} \]

Where:
- \( Ȳ_0 \) and \( Ȳ_1 \) are the means of the groups (given as 937.5 and 455 respectively).
- \( s_Y \) is the standard deviation of the entire sample (given as 217).
- \( ρ_0 \) and \( ρ_1 \) are the proportions of the total sample in each group (given as 0.2 and 0.8 respectively).

Given dataTo compute the Point Bi-serial correlation coefficient, we use the following formula:

\[ r_{pb} = \frac{Ȳ_1 - Ȳ_0}{s_Y} \sqrt{\frac{ρ_0ρ_1}{n}} \]

Where:
- \( Ȳ_0 \) and \( Ȳ_1 \) are the means of the groups (given as 937.5 and 455 respectively).
- \( s_Y \) is the standard deviation of the entire sample (given as 217).
- \( ρ_0 \) and \( ρ_1 \) are the proportions of the total sample in each group (given as 0.2 and 0.8 respectively).

Given 
\[ Ȳ_0 = 937.5 \]
\[ Ȳ_1 = 455 \]
\[ s_Y = 217 \]
\[ ρ_0 = 0.2 \]
\[ ρ_1 = 0.8 \]

Plugging these values into the formula:

\[ r_{pb} = \frac{455 - 937.5}{217} \sqrt{\frac{0.2 \cdot 0.8}{1}} \]

First, calculate the numerator:

\[ 455 - 937.5 = -482.5 \]

Next, calculate the fraction inside the square root:

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

Then, calculate the square root:

\[ \sqrt{0.16} = 0.4 \]

Now, combine these values:

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

\[ r_{pb} = -2.223 \cdot 0.4 \]

\[ r_{pb} = -0.8892 \]

Therefore, the Point Bi-serial correlation coefficient \( r_{pb} \) is approximately \( -0.889 \).

Would you like more details or have any questions?

Here are 5 related questions you can ask:
1. How does the Point Bi-serial correlation coefficient differ from Pearson's correlation coefficient?
2. What are the assumptions underlying the use of the Point Bi-serial correlation coefficient?
3. How do you interpret the value of the Point Bi-serial correlation coefficient?
4. Can the Point Bi-serial correlation coefficient be used for more than two groups?
5. How would the formula change if the proportions \( \rho_0 \) and \( \rho_1 \) are different?

Tip: The Point Bi-serial correlation coefficient is particularly useful when you need to assess the relationship between a continuous variable and a binary categorical variable.

Learn how to compute the Point Biserial Correlation Coefficient step-by-step with detailed explanations and calculations. Given Ȳ₀ = 937.5, Ȳ₁ = 455, s_Y = 217, ρ₀ = 0.2, ρ₁ = 0.8. Follow along to understand how each parameter affects the calculation of the Point Biserial Correlation Coefficient.

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