For a multiple regression model with 2 independent variables, R.sq = 0.904 and adjusted R. sq = 0.88, determine the number of observations (n) 6 7 9 10

1. Break Down the Problem

To find the number of observations (n) in a multiple regression model, we can use the relationship between R-squared (R²), adjusted R-squared (adjusted R²), and the number of predictors (p) and observations (n). The formula for adjusted R-squared is given by:

$\text{Adjusted } R^2 = 1 - \frac{(1 - R^2)(n - 1)}{n - p - 1}$

Where:

• $R^2$ is the R-squared value
• adjusted $R^2$ is the adjusted R-squared value
• $n$ is the number of observations
• $p$ is the number of predictors

In this case, $R^2 = 0.904$, adjusted $R^2 = 0.88$, and $p = 2$.

2. Relevant Concepts

To find $n$, rearrange the adjusted R-squared formula:

$0.88 = 1 - \frac{(1 - 0.904)(n - 1)}{n - 2 - 1}$

3. Analysis and Detail

Substituting the values into the equation:

1. Start with the equation: $0.88 = 1 - \frac{(1 - 0.904)(n - 1)}{n - 3}$

2. Simplify $1 - 0.904$: $1 - 0.904 = 0.096$

3. Thus, the equation becomes: $0.88 = 1 - \frac{0.096(n - 1)}{n - 3}$

4. Rearranging gives: $\frac{0.096(n - 1)}{n - 3} = 1 - 0.88 = 0.12$

5. Cross-multiplying the equation: $0.096(n - 1) = 0.12(n - 3)$

6. Expanding both sides: $0.096n - 0.096 = 0.12n - 0.36$

7. Rearranging to solve for $n$: $0.36 - 0.096 = 0.12n - 0.096n$ $0.264 = 0.024n$ $n = \frac{0.264}{0.024} = 11$

4. Verify and Summarize

After performing the calculations, we find that the number of observations $n$ is 11. However, this is not present in the provided options (6, 7, 9, 10), which might suggest a need for reconsideration of parameters or options provided.

Final Answer

The calculated number of observations is $n = 11$, but since this does not align with the options given (6, 7, 9, 10), it could be necessary to reassess the problem's parameters. However, based on the computations, if forced to choose from the listed options, we would have to indicate that none are correct.