Multiple Correlation Coefficient Properties
The sample multiple correlation coefficient, denoted here as $r$, measures the strength of the linear relationship between a dependent variable ($X_1$) and a set of predictor variables (e.g., $\{X_2, X_3\}$ or $\{X_2, X_3, X_4\}$).
Key properties of the multiple correlation coefficient ($R$) include:
- Non-negativity: The multiple correlation coefficient is always non-negative and cannot exceed 1. Mathematically, $0 \le r \le 1$.
- Monotonicity: Adding more predictor variables to the model can only increase or maintain the multiple correlation coefficient; it cannot decrease it. Therefore, when comparing $r_{1.23}$ (correlation with $X_2, X_3$) and $r_{1.234}$ (correlation with $X_2, X_3, X_4$), it must be that $r_{1.234} \ge r_{1.23}$.
Analyzing Coefficient Possibilities
Let's evaluate the given options based on these two properties:
- Option 1: $r_{1.23} = -0.3, r_{1.234} = 0.7$. This is impossible because multiple correlation coefficients cannot be negative (violates the non-negativity property).
- Option 2: $r_{1.23} = 0.7, r_{1.234} = 0.3$. This is impossible because adding variable $X_4$ decreased the correlation coefficient ($0.3 < 0.7$), violating the monotonicity property.
- Option 3: $r_{1.23} = 0.3, r_{1.234} = 0.7$. This is possible. Both values are within the range $[0, 1]$, and the coefficient increased ($0.7 \ge 0.3$) when an additional variable ($X_4$) was included.
- Option 4: $r_{1.23} = 0.7, r_{1.234} = -0.3$. This is impossible because the multiple correlation coefficient cannot be negative (violates the non-negativity property).
Conclusion
Based on the properties that multiple correlation coefficients must be between 0 and 1, and that they cannot decrease when more variables are added, only Option 3 presents a statistically possible scenario.
Important Questions from Ratio And Regression
- Consider the problem of drawing a sample of size 2 from a finite population of size 20. The sampling is done with replacement using probability proportional to size sampling scheme. The normed size measures $p_1, \cdots, p_{20}$ are given by $p_i = \frac{1}{40}$, $i = 1, \cdots, 10, \; p_i = \frac{3}{40}$, $i = 11, \cdots, 20$. The expected number of distinct units drawn is
- Consider a finite population of size $N$. Let $T_1$ be the sample mean based on a sample of size $n$ under simple random sampling with replacement (SRSWR) scheme. Let $T_2$ be the sample mean based on a stratified random sample of size $n$ where the samples are drawn from each of 4 strata using SRSWR scheme under proportional allocation. Then which of the following are sufficient conditions for $\text{Var}(T_1) = \text{Var}(T_2)$ to hold?
Suppose there are $k$ strata of $N = kM$ units each with size $M$. Draw a sample of size $n_i$ with replacement from the $i^{\text{th}}$ stratum and denote by $\bar{y}_i$ the sample mean of the study variable selected in the $i^{\text{th}}$ stratum, $i = 1, 2, \dots, k$. Define
$$ \bar{y}_s = \frac{1}{k}\sum_{i=1}^k \bar{y}_i \text{ and } \bar{y}_w = \frac{\sum_{i=1}^k n_i \bar{y}_i}{n} $$
Which of the following is necessarily true?- Suppose we draw a random sample of size $n$ from a population of size $N$, where $1 < n < N$, using simple random sampling without replacement scheme. Let $P$ be the population proportion of units possessing a particular attribute and $p$ be the corresponding sample proportion. Which of the following is an unbiased estimator for $P(1 - P)$?
Suppose there are $k$ groups each consisting of $N$ boys. We want to estimate the mean age $\mu$ of these $kN$ boys. Fix $1 < n < N$ and consider the following two sampling schemes.
I. Draw a simple random sample without replacement of size $kn$ out of all $kN$ boys.
II. From each of the $k$ groups draw a simple random sample with replacement of size $n$.
Let $\bar{Y}$ and $\bar{Y}_G$ be the respective sample mean ages for the two schemes. Which of the following are true?
