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
- A simple random sample (without replacement) of size $n$ is drawn from a finite population of size $N (\ge 7)$. What is the probability that the $4^{\text{th}}$ population unit is included in the sample but the $6^{\text{th}}$ population unit is not included in the sample?
- For a data set $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ the following two models were fitted using least square method.
Model 1: $y_i = \beta_0 + \beta_1 x_i \quad i = 1, 2, \dots n$
Model 2: $y_i = \beta_0^* + \beta_1^* x_i + \beta_2^* x_i^2 \quad i = 1, 2, \dots n$
Let $\hat{\beta}_0, \hat{\beta}_1$ be least square estimates of $\beta_0, \beta_1$ from model 1 and $\hat{\beta}_0^*, \hat{\beta}_1^*, \hat{\beta}_2^*$ be the least square estimates from model 2.
Let $A = \sum_1^n \left(y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_i)\right)^2$,
$B = \sum_1^n \left(y_i - (\hat{\beta}_0^* + \hat{\beta}_1^* x_i + \hat{\beta}_2^* x_i^2)\right)^2$
Then - Suppose $\bar{Y}$ is the sample mean of the study variables corresponding to a sample of size n using simple random sampling with replacement scheme and $\bar{Y}_{st}$ is the sample mean of the study variables corresponding to a sample of size n using stratified random sampling with replacement scheme under proportional allocation. Which of the following is/are sufficient condition/conditions for $Var(\bar{Y}) = Var(\bar{Y}_{st})$?
- 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?
