Question

Happiness and Being in a Relationship Let X, Y be two Bernoulli random variables and let p = P(X = 1) (the probability that X = 1) q = P(Y = 1) (the probability that Y = 1) r = P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1),...,(Xn, Yn) be a sample of n i.i.d. copies of (X,Y). Define and do not assumer = pq. = 3 X.Y.. We would like to know whether the facts of being happy and being in a relationship are independent of each other. In a given population, 1000 people (aged at least 21 years old) are sampled randomly with replacement and asked two questions: "Do you consider yourself as happy?" and "Are you involved in a relationship ?". The answers are summarized in the following table: Happy Not happy 205 179 In a relationship Not in a relationship 301 315 Denote by p, and r the true proportions or people that are happy, in a relationship, and both happy and in a relationship, respectively. Compute the values of p, and f. p =

Answer 1

Let:

$\small \begin{align*} X_i &= \begin{cases} 1 & \text{ if ith individual is happy} \\ 0 & \text{ if ith individual is not happy} \end{cases} \\ Y_i &= \begin{cases} 1 & \text{ if ith individual is in a relationship} \\ 0 & \text{ if ith individual is not in a relationship} \end{cases} \end{align*}$

Then,

$\small \begin{align*} \textbf{Sample proportion of } & \\ \textbf{people that are happy, } \boldsymbol{\hat{p}} &= \frac{1}{n} \sum_{i=1}^{n} {X_i} \ \ \ \ \ \text{[where n is the sample size]} \\ &= \frac{\text{No. of people that are happy}}{\text{Total no. of people}} \\ \text{[Using the} &\text{ definition of } X_i] \\ &= \frac{205+179}{205+179+301+315} \\ &= \frac{384}{1000} \\&= \bf 0.384 \ \ \ \ \ [ANSWER] \end{align*}$.

Also,

Sample proportion of people [where n is the sample size] that are in a relationship, i=1 No. of people that are in a relationship Total no. of people [Using the definition of Y] 205 + 301 205 + 179 +301 + 315 506 1000 [ANSWER] 0.506

Moreover,

$\small \begin{align*} \textbf{Sample proportion of people that are} & \\ \textbf{both happy and in a relationship, } \boldsymbol{\hat{r}} &= \frac{1}{n} \sum_{i=1}^{n} {X_iY_i} \ \ \ \ \ \text{[where n is the sample size]} \\ &= \frac{\text{No. of people that are both happy and in a relationship}}{\text{Total no. of people}} \\ \text{[Using the definition of } &X_i \text{ and }Y_i] \\ &= \frac{205}{205+179+301+315} \\ &= \frac{205}{1000} \\&= \bf 0.205 \ \ \ \ \ [ANSWER] \end{align*}$

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