Regression Slopes Relationship
The question involves two regression lines fitted using Ordinary Least Squares (OLS) for bivariate data $(y_i, x_i)$.
Analyst A fits $Y$ on $X$: $\hat{Y}_i = \hat{\alpha}_0 + \hat{\alpha}_1 x_i$. The slope is $\hat{\alpha}_1$.
Analyst B fits $X$ on $Y$: $\hat{X}_i = \hat{\beta}_0 + \hat{\beta}_1 y_i$. The slope is $\hat{\beta}_1$.
The slope coefficient for the regression of $Y$ on $X$ is given by $\hat{\alpha}_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$.
The slope coefficient for the regression of $X$ on $Y$ is given by $\hat{\beta}_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(Y)}$.
Key Condition Derivation
Let $r$ be the correlation coefficient between $X$ and $Y$, and let $s_x, s_y$ be their respective standard deviations. The slopes can be expressed as:
$\hat{\alpha}_1 = r \frac{s_y}{s_x}$
$\hat{\beta}_1 = r \frac{s_x}{s_y}$
The product of these slopes is:
$ \hat{\alpha}_1 \times \hat{\beta}_1 = \left( r \frac{s_y}{s_x} \right) \times \left( r \frac{s_x}{s_y} \right) = r^2 $
Since $r^2$ must be between 0 and 1 (inclusive), i.e., $0 \le r^2 \le 1$, the product of the slopes must also satisfy this condition:
$ 0 \le \hat{\alpha}_1 \times \hat{\beta}_1 \le 1 $
Analysis of Options
We examine the product $\hat{\alpha}_1 \times \hat{\beta}_1$ for each option:
- Option 1: $(-0.5, 2.5)$
- Product: $(-0.5) \times (2.5) = -1.25$
- Check: $-1.25 < 0$. Invalid.
- Option 2: $(0.5, 2.5)$
- Product: $(0.5) \times (2.5) = 1.25$
- Check: $1.25 > 1$. Invalid.
- Option 3: $(-2.0, 0.4)$
- Product: $(-2.0) \times (0.4) = -0.8$
- Check: $-0.8 < 0$. Invalid.
- Option 4: $(-2.0, -0.4)$
- Product: $(-2.0) \times (-0.4) = 0.8$
- Check: $0 \le 0.8 \le 1$. Valid.
Conclusion
The only pair that satisfies the condition $0 \le \hat{\alpha}_1 \times \hat{\beta}_1 \le 1$ is $(-2.0, -0.4)$.
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?
