# Given Information - \( X \) is a random variable with a Bernoulli distribution: \( X \sim \mathrm{Ber}(p) \). - For each \( t \in \mathbb{R} \) with \( t \geq \), define \[ X_t = \begin{cases} \cos(\pi t) & \text{if } X = \\ \sin(\pi t) & \text{if } X = 1 \end{cases} \] # What to Find (a) Draw all the trajectories of the process \(\{X_t : t \geq \}\). (b) Compute the distribution of the random variable \( X_t \). (c) Compute \( \mathbb{E}(X_t) \). (d) Determine \( \mathbb{E}(X_t \mid X_) \). # Definitions / Concepts Used - **Bernoulli Distribution:** \( X \sim \mathrm{Ber}(p) \) means \( P(X=1) = p \), \( P(X=) = 1-p \). - **Random Process:** \( X_t \) is defined in terms of \( X \), so the entire process is determined by the value of \( X \). - **Expectation:** \( \mathbb{E}(Y) = \sum_y yP(Y=y) \) for discrete \( Y \). - **Conditional Expectation:** \( \mathbb{E}(Y|Z) \) is the expectation of \( Y \) given \( Z \). --- # Solution **(a) All Trajectories of the Process \(\{ X_t : t \geq \}\)** There are only two possible trajectories, depending on the value of \( X \): - If \( X= \): \( X_t = \cos(\pi t) \) for all \( t \geq \). - If \( X=1 \): \( X_t = \sin(\pi t) \) for all \( t \geq \). So, the possible trajectories are: - The curve \( t \mapsto \cos(\pi t) \). - The curve \( t \mapsto \sin(\pi t) \). --- **(b) Distribution of \( X_t \)** Since \( X_t \) is determined by \( X \): - \( P(X_t = \cos(\pi t)) = P(X=) = 1-p \). - \( P(X_t = \sin(\pi t)) = P(X=1) = p \). Thus, for each \( t \), \( X_t \) takes value \( \cos(\pi t) \) with probability \( 1-p \), and \( \sin(\pi t) \) with probability \( p \). --- **(c) Expected Value \( \mathbb{E}(X_t) \)** \[ \mathbb{E}(X_t) = \cos(\pi t) \cdot (1-p) + \sin(\pi t) \cdot p \] --- **(d) Conditional Expectation \( \mathbb{E}(X_t | X_) \)** \( X_ \) is either \( \cos() = 1 \) if \( X= \), or \( \sin() = \) if \( X=1 \). - If \( X_ = 1 \), then \( X= \), so \( X_t = \cos(\pi t) \). - If \( X_ = \), then \( X=1 \), so \( X_t = \sin(\pi t) \). Thus, \[ \mathbb{E}(X_t | X_) = \begin{cases} \cos(\pi t) & \text{if } X_ = 1 \\ \sin(\pi t) & \text{if } X_ = \end{cases} \] --- # Summary - The process has two trajectories: \( \cos(\pi t) \) and \( \sin(\pi t) \). - \( X_t \) is \( \cos(\pi t) \) with probability \( 1-p \), and \( \sin(\pi t) \) with probability \( p \). - \( \mathbb{E}(X_t) = \cos(\pi t)(1-p) + \sin(\pi t)p \). - \( \mathbb{E}(X_t|X_) = \cos(\pi t) \) if \( X_=1 \); \( \sin(\pi t) \) if \( X_= \).

# Given Information - \( X \) is a random variable with a Bernoulli distribution: \( X \sim \mathrm{Ber}(p) \). - For each \( t \in \mathbb{R} \) with \( t \geq 0 \), define: \[ X_t = \begin{cases} \cos(\pi t) & \text{if } X = 0 \\ \sin(\pi t) & \text{if } X = 1 \end{cases} \] # What to Find - Compute \( \mathbb{E}(X_T | X_0) \). # Definitions / Concepts Used - **Bernoulli Distribution:** \( X \sim \mathrm{Ber}(p) \) means \( P(X=1) = p \), \( P(X=0) = 1-p \). - **Conditional Expectation:** \( \mathbb{E}(Y|Z) \) is the expectation of \( Y \) given \( Z \). - **Random Process:** \( X_t \) is defined in terms of \( X \), thus its value depends on \( X \). --- # Solution To calculate \( \mathbb{E}(X_T | X_0) \), we need to consider the possible values of \( X_0 \): 1. **If \( X_0 = 0 \)**: - Then \( X = 0 \), leading to \( X_T = \cos(\pi T) \). - Thus, \[ \mathbb{E}(X_T | X_0 = 0) = \cos(\pi T). \] 2. **If \( X_0 = 1 \)**: - Then \( X = 1 \), leading to \( X_T = \sin(\pi T) \). - Thus, \[ \mathbb{E}(X_T | X_0 = 1) = \sin(\pi T). \] The conditional expectation can now be expressed as: \[ \mathbb{E}(X_T | X_0) = \begin{cases} \cos(\pi T) & \text{if } X_0 = 0 \\ \sin(\pi T) & \text{if } X_0 = 1 \end{cases} \] --- # Summary - \( \mathbb{E}(X_T | X_0) = \cos(\pi T) \) if \( X_0 = 0 \); \( \sin(\pi T) \) if \( X_0 = 1 \).

# Given Information - \( X \) is a random variable with a Bernoulli distribution: \( X \sim \mathrm{Ber}(p) \). - For each \( t \in \mathbb{R} \) with \( t \geq 0 \), define: \[ X_t = \begin{cases} \cos(\pi t) & \text{if } X = 0 \\ \sin(\pi t) & \text{if } X = 1 \end{cases} \] # What to Find (a) Draw all the trajectories of the process \( \{X_t : t \geq 0\} \). (b) Compute the distribution of the random variable \( X_t \). (c) Compute \( \mathbb{E}(X_t) \). (d) Determine \( \mathbb{E}(X_T | X_0) \). # Definitions / Concepts Used - **Bernoulli Distribution:** \( X \sim \mathrm{Ber}(p) \) means \( P(X=1) = p \), \( P(X=0) = 1-p \). - **Random Process:** \( X_t \) is defined in terms of \( X \), so the entire process is determined by the value of \( X \). - **Expectation:** \( \mathbb{E}(Y) = \sum_y y P(Y=y) \) for discrete \( Y \). - **Conditional Expectation:** \( \mathbb{E}(Y|Z) \) is the expectation of \( Y \) given \( Z \). --- # Solution **(a) All Trajectories of the Process \(\{ X_t : t \geq 0\}\)** There are two possible trajectories based on the value of \( X \): - If \( X=0 \): \( X_t = \cos(\pi t) \) for all \( t \geq 0 \). - If \( X=1 \): \( X_t = \sin(\pi t) \) for all \( t \geq 0 \). Thus, the possible trajectories are: - The curve \( t \mapsto \cos(\pi t) \). - The curve \( t \mapsto \sin(\pi t) \). --- **(b) Distribution of \( X_t \)** Since \( X_t \) is determined by \( X \): - \( P(X_t = \cos(\pi t)) = P(X=0) = 1-p \). - \( P(X_t = \sin(\pi t)) = P(X=1) = p \). Thus, for each \( t \): - \( X_t \) takes the value \( \cos(\pi t) \) with probability \( 1-p \). - \( X_t \) takes the value \( \sin(\pi t) \) with probability \( p \). --- **(c) Expected Value \( \mathbb{E}(X_t) \)** Calculating the expected value: \[ \mathbb{E}(X_t) = \cos(\pi t) \cdot (1-p) + \sin(\pi t) \cdot p \] --- **(d) Conditional Expectation \( \mathbb{E}(X_T | X_0) \)** To compute \( \mathbb{E}(X_T | X_0) \), we consider the values of \( X_0 \): 1. **If \( X_0 = 0 \)**: - Then \( X = 0 \), leading to \( X_T = \cos(\pi T) \). - Therefore, \[ \mathbb{E}(X_T | X_0 = 0) = \cos(\pi T). \] 2. **If \( X_0 = 1 \)**: - Then \( X = 1 \), leading to \( X_T = \sin(\pi T) \). - Thus, \[ \mathbb{E}(X_T | X_0 = 1) = \sin(\pi T). \] The overall conditional expectation can be expressed as: \[ \mathbb{E}(X_T | X_0) = \begin{cases} \cos(\pi T) & \text{if } X_0 = 0 \\ \sin(\pi T) & \text{if } X_0 = 1 \end{cases} \] --- # Summary - The process has two trajectories: \( \cos(\pi t) \) and \( \sin(\pi t) \). - \( X_t \) is \( \cos(\pi t) \) with probability \( 1-p \), and \( \sin(\pi t) \) with probability \( p \). - \( \mathbb{E}(X_t) = \cos(\pi t)(1-p) + \sin(\pi t)p \). - \( \mathbb{E}(X_T | X_0) = \cos(\pi T) \) if \( X_0 = 0 \); \( \sin(\pi T) \) if \( X_0 = 1 \).

# Given Information - The weather in Oz has specific rules: - No two consecutive sunny days. - A sunny day is followed by either a rainy or snowy day (each with equal probability). - On a bad weather day (rainy or snowy), the weather can either stay the same or switch, with the following probabilities: - If it's raining, it can rain again (with probability \( \frac{1}{2} \)) or switch to sunny (with probability \( \frac{1}{4} \)) or snow (with probability \( \frac{1}{4} \)). - If it's snowing, the transitions are similar. # What to Find (a) Define a state space. (b) Specify the transition probabilities. (c) Indicate a possible initial distribution. (d) Argue why it is a Markov chain. # Definitions / Concepts Used - **Markov Chain:** A stochastic process where the future state depends only on the current state and not on the sequence of events that preceded it. - **State Space:** The set of all possible states in a Markov process. - **Transition Probability:** The probability of moving from one state to another. --- # Solution **(a) State Space** Define the state space \( S \) as follows: - \( S = \{ S, R, W \} \) - \( S \): Sunny - \( R \): Rainy - \( W \): Snowy **(b) Transition Probabilities** Based on the rules given: - From \( S \) (Sunny): - \( P(S \to R) = \frac{1}{2} \) - \( P(S \to W) = \frac{1}{2} \) - From \( R \) (Rainy): - \( P(R \to R) = \frac{1}{2} \) - \( P(R \to S) = \frac{1}{4} \) - \( P(R \to W) = \frac{1}{4} \) - From \( W \) (Snowy): - \( P(W \to W) = \frac{1}{2} \) - \( P(W \to S) = \frac{1}{4} \) - \( P(W \to R) = \frac{1}{4} \) **Transition Probability Matrix \( P \)**: \[ P = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2} \end{bmatrix} \] Where rows represent the current state and columns represent the next state: - Row 1: \( S \) - Row 2: \( R \) - Row 3: \( W \) **(c) Initial Distribution** A possible initial distribution could be: \[ \pi = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \] This indicates that the weather starts sunny with probability 1. **(d) Why it is a Markov Chain** The weather on any given day depends only on the weather of the previous day and not on the sequence of days before that. This memoryless property fulfills the Markov property, confirming that the weather model can be represented as a Markov chain. --- # Summary - **State Space:** \( S = \{ S, R, W \} \) - **Transition Probabilities:** - From \( S \): \( P(S \to R) = \frac{1}{2}, P(S \to W) = \frac{1}{2} \) - From \( R \): \( P(R \to R) = \frac{1}{2}, P(R \to S) = \frac{1}{4}, P(R \to W) = \frac{1}{4} \) - From \( W \): \( P(W \to W) = \frac{1}{2}, P(W \to S) = \frac{1}{4}, P(W \to R) = \frac{1}{4} \) - **Initial Distribution:** \( \pi = [1, 0, 0] \) - **Markov Chain Property:** The weather on any day depends only on the previous day's weather.

# Given Information - The weather in Oz has specific rules: - No two consecutive sunny days. - A sunny day is followed by either a rainy or snowy day (each with equal probability). - On a bad weather day (rainy or snowy), the weather can either stay the same or switch, with the following probabilities: - If it's raining, it can rain again (with probability \( \frac{1}{2} \)) or switch to sunny (with probability \( \frac{1}{4} \)) or snow (with probability \( \frac{1}{4} \)). - If it's snowing, the transitions are similar. # What to Find (a) Define a state space. (b) Specify the transition probabilities. (c) Indicate a possible initial distribution. (d) Argue why it is a Markov chain. # Definitions / Concepts Used - **Markov Chain:** A stochastic process where the future state depends only on the current state and not on the sequence of events that preceded it. - **State Space:** The set of all possible states in a Markov process. - **Transition Probability:** The probability of moving from one state to another. --- # Solution **(a) State Space** Define the state space \( S \) as follows: - \( S = \{ S, R, W \} \) - \( S \): Sunny - \( R \): Rainy - \( W \): Snowy **(b) Transition Probabilities** Based on the rules given: - From \( S \) (Sunny): - \( P(S \to R) = \frac{1}{2} \) - \( P(S \to W) = \frac{1}{2} \) - From \( R \) (Rainy): - \( P(R \to R) = \frac{1}{2} \) - \( P(R \to S) = \frac{1}{4} \) - \( P(R \to W) = \frac{1}{4} \) - From \( W \) (Snowy): - \( P(W \to W) = \frac{1}{2} \) - \( P(W \to S) = \frac{1}{4} \) - \( P(W \to R) = \frac{1}{4} \) **Transition Probability Matrix \( P \)**: \[ P = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2} \end{bmatrix} \] Where rows represent the current state and columns represent the next state: - Row 1: \( S \) - Row 2: \( R \) - Row 3: \( W \) **(c) Initial Distribution** A possible initial distribution could be: \[ \pi = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \] This indicates that the weather starts sunny with probability 1. **(d) Why it is a Markov Chain** The weather on any given day depends only on the weather of the previous day and not on the sequence of days before that. This memoryless property fulfills the Markov property, confirming that the weather model can be represented as a Markov chain. --- # Summary - **State Space:** \( S = \{ S, R, W \} \) - **Transition Probabilities:** - From \( S \): \( P(S \to R) = \frac{1}{2}, P(S \to W) = \frac{1}{2} \) - From \( R \): \( P(R \to R) = \frac{1}{2}, P(R \to S) = \frac{1}{4}, P(R \to W) = \frac{1}{4} \) - From \( W \): \( P(W \to W) = \frac{1}{2}, P(W \to S) = \frac{1}{4}, P(W \to R) = \frac{1}{4} \) - **Initial Distribution:** \( \pi = [1, 0, 0] \) - **Markov Chain Property:** The weather on any day depends only on the previous day's weather.

# Given Information - Two-thirds of people encountered in the National Park of Atlantis are foreign tourists. - Foreign tourists provide correct directions with a probability of \( \frac{3}{4} \). - Indigenous Atlanteans always provide false answers. - You cannot distinguish between tourists and Atlanteans. - The probability that the correct exit is East is \( \epsilon \). # What to Find (a) The updated belief that East is correct after receiving an answer of "West." (b) The updated belief that East is correct after receiving the answer "West" twice. (c) The updated belief of the probability that the passer-by is a foreign tourist after receiving the answer "West" once. # Definitions / Concepts Used - **Bayes' Theorem:** \[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \] - **Prior Probability:** The initial belief about an event before considering new evidence. - **Likelihood:** The probability of receiving the evidence given a particular hypothesis. --- # Solution **(a) Updated Belief after First Answer** Define the events: - \( A \): "East is correct." - \( B \): "The passer-by answers West." Using Bayes' Theorem: \[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \] 1. **Calculate Prior Probabilities:** - \( P(A) = \epsilon \) - \( P(A^c) = 1 - \epsilon \) 2. **Calculate Likelihoods:** - If East is correct (\( A \)), then the passer-by (foreign tourist) answers West incorrectly with probability \( \frac{1}{4} \): \[ P(B | A) = \frac{1}{3} \cdot \frac{1}{4} + \frac{2}{3} \cdot 1 = \frac{1}{12} + \frac{2}{3} = \frac{1 + 8}{12} = \frac{9}{12} = \frac{3}{4} \] - If East is not correct (\( A^c \)), the passer-by answers West (either tourist or Atlantean) with probability \( \frac{2}{3} \cdot \frac{3}{4} + \frac{1}{3} \cdot 1 = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \): 3. **Calculate Overall Probability \( P(B) \):** \[ P(B) = P(B | A) P(A) + P(B | A^c) P(A^c) = \frac{3}{4} \epsilon + \frac{5}{6} (1 - \epsilon) \] Now substitute back into Bayes' Theorem: \[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} = \frac{\frac{3}{4} \epsilon}{\frac{3}{4} \epsilon + \frac{5}{6} (1 - \epsilon)} \] **(b) Updated Belief after Second Answer (West)** Repeat the same process using the updated \( P(A | B) \) as the prior for the second question: 1. The likelihoods remain the same because the answer is independent. 2. Calculate \( P(B | A) \) and \( P(B | A^c) \) again. 3. Calculate \( P(B) \) again with the updated prior. The final expression will have similar structure but with updated prior probabilities. **(c) Updated Belief that the Passer-by is a Foreign Tourist** We need to calculate \( P(T | B) \) where \( T \) is the event that the passer-by is a foreign tourist: \[ P(T | B) = \frac{P(B | T) P(T)}{P(B)} \] 1. **Calculate Prior Probabilities:** - \( P(T) = \frac{2}{3} \) - \( P(A) = \epsilon \) 2. **Calculate Likelihoods:** - If the passer-by is a tourist: \[ P(B | T) = \frac{1}{4} \] - If the passer-by is an Atlantean: \[ P(B | A^c) = 1 \] 3. **Calculate Overall Probability \( P(B) \) again** using: \[ P(B) = P(B | T) P(T) + P(B | A^c) P(A^c) \] Finally, substitute into Bayes' Theorem: \[ P(T | B) = \frac{P(B | T) P(T)}{P(B)} \] --- # Summary - (a) Updated belief that East is correct after one answer of "West": \[ P(A | B) = \frac{\frac{3}{4} \epsilon}{\frac{3}{4} \epsilon + \frac{5}{6} (1 - \epsilon)} \] - (b) Updated belief after two answers of "West" will follow similar reasoning but with the new prior. - (c) Updated belief of the passer-by being a foreign tourist after one answer of "West": \[ P(T | B) = \frac{P(B | T) P(T)}{P(B)} = \frac{\frac{1}{4} \cdot \frac{2}{3}}{P(B)} \] where \( P(B) \) is calculated as mentioned.

# Given Information - Two-thirds of people in the National Park of Atlantis are foreign tourists. - Foreign tourists provide correct directions with a probability of \( \frac{3}{4} \). - Indigenous Atlanteans always provide false answers. - You cannot distinguish between tourists and Atlanteans. - The probability that the correct exit is East is \( \epsilon \). # What to Find (a) The updated belief that East is correct after receiving an answer of "West." (b) The updated belief that East is correct after receiving the answer "West" twice. (c) The updated belief of the probability that the passer-by is a foreign tourist after receiving the answer "West" once. # Definitions / Concepts Used - **Bayes' Theorem:** \[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \] - **Prior Probability:** The initial belief about an event before considering new evidence. - **Likelihood:** The probability of receiving the evidence given a particular hypothesis. --- # Solution **(a) Updated Belief after First Answer** Define the events: - \( A \): "East is correct." - \( B \): "The passer-by answers West." Using Bayes' Theorem: \[ P(A | B) = \frac{P(B | A) P(A)}{P(B)} \] 1. **Prior Probabilities:** - \( P(A) = \epsilon \) - \( P(A^c) = 1 - \epsilon \) 2. **Likelihoods:** - If East is correct (\( A \)): - \( P(B | A) = \frac{1}{4} \) (incorrect answer from tourist) - If East is not correct (\( A^c \)): - \( P(B | A^c) = \frac{2}{3} \cdot \frac{1}{4} + \frac{1}{3} = \frac{1}{6} + \frac{1}{3} = \frac{3}{6} = \frac{1}{2} \) 3. **Overall Probability \( P(B) \):** \[ P(B) = P(B | A) P(A) + P(B | A^c) P(A^c) = \frac{1}{4} \epsilon + \frac{1}{2} (1 - \epsilon) \] Now substitute back into Bayes' Theorem: \[ P(A | B) = \frac{\frac{1}{4} \epsilon}{\frac{1}{4} \epsilon + \frac{1}{2} (1 - \epsilon)} \] **(b) Updated Belief after Second Answer (West)** Repeat the process using the updated \( P(A | B) \) as the prior: 1. The likelihoods remain the same. 2. Overall probability \( P(B) \) will be calculated using the same method: \[ P(B) = P(B | A) P(A) + P(B | A^c) P(A^c) \] Where: - \( P(B | A) = \frac{1}{4} \) - \( P(B | A^c) = \frac{1}{2} \) The final expression will now be: \[ P(A | B \cap B) = \frac{\frac{1}{4} P(A | B)}{\frac{1}{4} P(A | B) + \frac{1}{2} (1 - P(A | B))} \] **(c) Updated Belief that the Passer-by is a Foreign Tourist** Calculate \( P(T | B) \): \[ P(T | B) = \frac{P(B | T) P(T)}{P(B)} \] 1. **Prior Probabilities:** - \( P(T) = \frac{2}{3} \) - \( P(A) = \epsilon \) 2. **Likelihoods:** - For a tourist: \[ P(B | T) = \frac{1}{4} \] - For an Atlantean: \[ P(B | A^c) = 1 \] 3. **Overall Probability \( P(B) \)**: \[ P(B) = P(B | T) P(T) + P(B | A^c) P(A^c) = \frac{1}{4} \cdot \frac{2}{3} + 1 \cdot \frac{1}{3} = \frac{1}{6} + \frac{1}{3} = \frac{1 + 2}{6} = \frac{3}{6} = \frac{1}{2} \] Finally, substitute into Bayes' Theorem: \[ P(T | B) = \frac{\frac{1}{4} \cdot \frac{2}{3}}{\frac{1}{2}} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3} \] --- # Summary - (a) Updated belief that East is correct after one answer of "West": \[ P(A | B) = \frac{\frac{1}{4} \epsilon}{\frac{1}{4} \epsilon + \frac{1}{2} (1 - \epsilon)} \] - (b) Updated belief after two answers of "West" will similarly follow: \[ P(A | B \cap B) = \frac{\frac{1}{4} P(A | B)}{\frac{1}{4} P(A | B) + \frac{1}{2} (1 - P(A | B))} \] - (c) Updated belief of the passer-by being a foreign tourist after one answer of "West": \[ P(T | B) = \frac{\frac{1}{4} \cdot \frac{2}{3}}{\frac{1}{2}} = \frac{1}{3} \]

# Given Information - Regression equation: \[ \hat{y} = 12 + 3x \] - Coefficient of determination: \( R^2 = 0.81 \) # What to Find 1. Interpret the slope of the regression equation. 2. Determine the correlation coefficient. 3. Explain how the predicted \( y \) changes when \( x \) increases by 2 units. # Definitions / Concepts Used - **Slope**: The coefficient of \( x \) in the regression equation, representing the change in \( y \) for a one-unit increase in \( x \). - **Correlation Coefficient**: A measure of the strength and direction of the linear relationship between two variables, typically denoted as \( r \). - **Coefficient of Determination (\( R^2 \))**: Represents the proportion of the variance for the dependent variable that's explained by the independent variable(s) in the model. --- # Solution 1. **Interpret the Slope**: - The slope of the regression equation is \( 3 \). This means that for every one-unit increase in \( x \), the predicted value of \( y \) increases by \( 3 \) units. 2. **Correlation Coefficient**: - The correlation coefficient \( r \) can be derived from \( R^2 \): \[ r = \sqrt{R^2} = \sqrt{0.81} = 0.9 \] Since the slope is positive, the correlation coefficient is also positive: \[ r = 0.9 \] This indicates a strong positive linear relationship between \( x \) and \( y \). 3. **Change in Predicted \( y \) for \( x \) Increase**: - If \( x \) increases by \( 2 \) units, the change in the predicted \( y \) can be calculated as: \[ \text{Change in } \hat{y} = \text{slope} \times \text{change in } x = 3 \times 2 = 6 \] Therefore, the predicted \( y \) increases by \( 6 \) units when \( x \) increases by \( 2 \) units. --- # Summary - The slope of the regression equation indicates that for each one-unit increase in \( x \), the predicted \( y \) increases by \( 3 \) units. - The correlation coefficient is \( r = 0.9 \), indicating a strong positive linear relationship. - If \( x \) increases by \( 2 \) units, the predicted \( y \) increases by \( 6 \) units.

# Given Information - Regression equation: \[ \hat{y} = 12 + 3x \] - Coefficient of determination: \( R^2 = 0.81 \) # What to Find 1. Interpret the slope of the regression equation. 2. Calculate the correlation coefficient. 3. Determine the change in predicted \( y \) when \( x \) increases by 2 units. # Definitions / Concepts Used - **Slope**: The coefficient of \( x \) in the regression equation, indicating how much \( y \) changes for each one-unit increase in \( x \). - **Correlation Coefficient**: A statistic that measures the degree of linear relationship between two variables, denoted \( r \). - **Coefficient of Determination (\( R^2 \))**: The fraction of the variance in the dependent variable that is predictable from the independent variable. --- # Solution 1. **Interpret the Slope**: - The slope of the regression equation is \( 3 \). This means that for each additional unit increase in \( x \), the predicted value of \( y \) increases by \( 3 \) units. This indicates a direct positive relationship between \( x \) and \( y \). 2. **Calculate the Correlation Coefficient**: - The correlation coefficient \( r \) can be calculated from \( R^2 \): \[ r = \sqrt{R^2} = \sqrt{0.81} = 0.9 \] Since the slope of the regression line is positive, the correlation coefficient is also positive: \[ r = 0.9 \] This value indicates a strong positive correlation between the variables \( x \) and \( y \). 3. **Change in Predicted \( y \) for Increase in \( x \)**: - When \( x \) increases by \( 2 \) units, the change in the predicted \( y \) value is calculated as follows: \[ \text{Change in } \hat{y} = \text{slope} \times \text{change in } x = 3 \times 2 = 6 \] Thus, the predicted \( y \) increases by \( 6 \) units when \( x \) increases by \( 2 \) units. --- # Summary - The slope of the regression indicates that every one-unit increase in \( x \) results in a \( 3 \) unit increase in predicted \( y \). - The correlation coefficient is \( r = 0.9 \), reflecting a strong positive linear relationship. - A \( 2 \) unit increase in \( x \) results in a \( 6 \) unit increase in the predicted value of \( y \).

# Given Information - Regression equation: \[ \hat{y} = 12 + 3x \] - Coefficient of determination: \( R^2 = 0.81 \) # Objectives 1. Explain the meaning of the slope in the regression equation. 2. Calculate the correlation coefficient. 3. Analyze how the predicted \( y \) changes with an increase of \( 2 \) units in \( x \). # Definitions / Concepts Used - **Slope**: The value of the coefficient of \( x \) in the regression equation, indicating the amount by which \( y \) is expected to change for each unit increase in \( x \). - **Correlation Coefficient**: A statistic that quantifies the strength and direction of the linear relationship between two variables, represented as \( r \). - **Coefficient of Determination (\( R^2 \))**: A measure that explains how much of the variance in the dependent variable can be explained by the independent variable. --- # Solution 1. **Interpretation of the Slope**: - The slope of the regression line is \( 3 \). This indicates that for every one-unit increase in \( x \), the predicted value of \( y \) increases by \( 3 \) units. This reflects a positive relationship, meaning as \( x \) rises, \( y \) also tends to rise. 2. **Calculation of the Correlation Coefficient**: - The correlation coefficient \( r \) is derived from the \( R^2 \) value: \[ r = \sqrt{R^2} = \sqrt{0.81} = 0.9 \] Given that the slope is positive, the correlation coefficient is also positive: \[ r = 0.9 \] This indicates a strong positive correlation between \( x \) and \( y \). 3. **Effect of \( x \) Increasing by 2 Units**: - When \( x \) is increased by \( 2 \) units, the change in the predicted value of \( y \) is computed as follows: \[ \text{Change in } \hat{y} = \text{slope} \times \text{change in } x = 3 \times 2 = 6 \] Consequently, the predicted \( y \) will increase by \( 6 \) units as a result of a \( 2 \) unit increase in \( x \). --- # Summary - The slope of the regression equation indicates that each one-unit increment in \( x \) results in a \( 3 \) unit increase in the predicted \( y \). - The correlation coefficient is \( r = 0.9 \), which signifies a strong positive linear relationship between the variables. - An increase of \( 2 \) units in \( x \) leads to a \( 6 \) unit rise in the predicted value of \( y \).

# Given Information - The regression equation is given by: \[ \hat{y} = 12 + 3x \] - The coefficient of determination is \( R^2 = 0.81 \). # Goals 1. Interpret the slope in the context of the regression equation. 2. Calculate the correlation coefficient. 3. Explain how the predicted value of \( y \) changes with a \( 2 \) unit increase in \( x \). # Definitions / Concepts Used - **Slope**: The coefficient of \( x \) in the regression equation, which indicates the expected change in \( y \) for a one-unit increase in \( x \). - **Correlation Coefficient**: A numerical measure of the strength and direction of the linear relationship between two variables, denoted as \( r \). - **Coefficient of Determination (\( R^2 \))**: Represents the proportion of variance in the dependent variable that can be explained by the independent variable(s) in the regression model. --- # Solution 1. **Interpretation of the Slope**: - In the regression equation, the slope is \( 3 \). This implies that for each unit increase in \( x \), the predicted \( y \) increases by \( 3 \) units. This establishes a direct and positive relationship between the two variables, indicating that as \( x \) increases, \( y \) also increases. 2. **Calculation of the Correlation Coefficient**: - The correlation coefficient \( r \) can be calculated from \( R^2 \): \[ r = \sqrt{R^2} = \sqrt{0.81} = 0.9 \] Since the slope of the regression equation is positive, the correlation coefficient is also positive: \[ r = 0.9 \] This value indicates a strong positive correlation between \( x \) and \( y \). 3. **Impact of Increasing \( x \) by 2 Units**: - If \( x \) is increased by \( 2 \) units, the change in the predicted value of \( y \) can be calculated as follows: \[ \text{Change in } \hat{y} = \text{slope} \times \text{change in } x = 3 \times 2 = 6 \] Therefore, the predicted value of \( y \) increases by \( 6 \) units when \( x \) is increased by \( 2 \) units. --- # Summary - The slope of the regression equation indicates that each additional unit increase in \( x \) results in a \( 3 \) unit increase in the predicted value of \( y \). - The correlation coefficient is \( r = 0.9 \), reflecting a strong positive linear relationship. - An increase of \( 2 \) units in \( x \) results in a \( 6 \) unit increase in the predicted value of \( y \).

# Given Information - The regression equation is: \[ \hat{y} = 12 + 3x \] - The coefficient of determination, \( R^2 \), is \( 0.81 \). # Objectives 1. Explain the significance of the slope in the regression formula. 2. Calculate the correlation coefficient between \( x \) and \( y \). 3. Determine how the predicted value of \( y \) changes when \( x \) increases by 2 units. # Key Concepts - **Slope**: The coefficient of \( x \) in the regression equation, indicating the change in \( y \) for each unit increase in \( x \). - **Correlation Coefficient**: A statistic that measures the strength and direction of the linear relationship between two variables, denoted as \( r \). - **Coefficient of Determination (\( R^2 \))**: A metric that indicates the proportion of the variance in the dependent variable that can be explained by the independent variable. --- # Solution 1. **Significance of the Slope**: - The slope of the regression equation is \( 3 \). This means that for every one-unit increase in \( x \), the predicted value of \( y \) increases by \( 3 \) units. This reflects a positive correlation, indicating that as \( x \) increases, \( y \) also tends to increase. 2. **Correlation Coefficient Calculation**: - The correlation coefficient \( r \) is derived from the coefficient of determination \( R^2 \): \[ r = \sqrt{R^2} = \sqrt{0.81} = 0.9 \] Given that the slope of the regression is positive, the correlation coefficient is also positive: \[ r = 0.9 \] This indicates a strong positive relationship between \( x \) and \( y \). 3. **Effect of Increasing \( x \) by 2 Units**: - When \( x \) is increased by \( 2 \) units, the change in the predicted value of \( y \) can be calculated as: \[ \text{Change in } \hat{y} = \text{slope} \times \text{change in } x = 3 \times 2 = 6 \] Thus, the predicted \( y \) increases by \( 6 \) units when \( x \) increases by \( 2 \) units. --- # Summary - The slope of the regression indicates that each one-unit increase in \( x \) leads to a \( 3 \) unit increase in the predicted \( y \). - The correlation coefficient is \( r = 0.9 \), signifying a strong positive linear relationship between the variables. - An increase of \( 2 \) units in \( x \) results in a \( 6 \) unit increase in the predicted value of \( y \).