Confidence Interval Estimate for Population Mean

STEP 1 - Given Data

Since $\sigma$ is known, we use the Z-distribution.

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STEP 2 - Solution

Part (a): Standard Error of the Mean

$$SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{49}} = \frac{5}{7} = 0.7143$$

$$\boxed{SE \approx 0.714}$$

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Part (b): 95% Confidence Interval

Step 1: Z value

For 95% confidence:

• $\alpha = 0.05$, so $\alpha/2 = 0.025$
• $Z_{0.025} = 1.96$

Step 2: Margin of error

$$E = Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} = 1.96 \times 0.7143 = 1.4000$$

Step 3: Confidence interval

$$\text{Lower} = \bar{x} - E = 24.80 - 1.40 = 23.40$$

$$\text{Upper} = \bar{x} + E = 24.80 + 1.40 = 26.20$$

$$\boxed{23.40 \leq \mu \leq 26.20}$$

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Conclusion

• Standard error of the mean $= 0.714$
• 95% confidence interval for the population mean $$= $23.40$$ to $$$26.20$$

We are 95% confident that the true mean dinner amount per customer lies between $23.40 and $26.20.

Markov Chain and Its Characteristics

Definition of Stochastic Process

A stochastic process is a family of random variables indexed by time as a parameter. It is denoted as {X(t), t ∈ T}, where:

• T = index set (time parameter)
• X(t) = random variable at time t
• I = state space (the set of all possible values assumed by X(t))

The values assumed by the random variable X(t) are called states.

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Definition of Markov Chain

A Markov Chain is a stochastic process {X(t), t = 0, 1, 2, ...} that satisfies the Markov property (memoryless property):

$$P[X(t+1) = j \mid X(t) = i,\ X(t-1) = i_{t-1},\ \ldots,\ X(0) = i_0] = P[X(t+1) = j \mid X(t) = i]$$

That is, the future state depends only on the present state, not on the past states.

The probability $P[X(t+1) = j \mid X(t) = i] = p_{ij}$ is called the transition probability from state $i$ to state $j$.

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Characteristics of a Markov Chain

1. Markov Property (Memoryless Property)

The conditional probability of any future state depends only on the current state, not on the sequence of states that preceded it. This is the defining characteristic.

2. Transition Probability Matrix (TPM)

The one-step transition probabilities are arranged in a matrix $P = [p_{ij}]$, where:

• $p_{ij} \geq 0$ for all $i, j$
• $\sum_{j} p_{ij} = 1$ for each row $i$ (rows sum to 1)

Such a matrix is called a stochastic matrix.

3. Homogeneity (Time Homogeneous)

A Markov chain is said to be homogeneous if the transition probabilities do not depend on time $t$, i.e., $p_{ij}$ remains constant over time.

4. Chapman-Kolmogorov Equation

The $n$-step transition probability satisfies:

$$p_{ij}^{(n)} = \sum_{k} p_{ik}^{(r)} \cdot p_{kj}^{(n-r)}, \quad 0 < r < n$$

In matrix form: $P^{(n)} = P^n$

5. Steady State (Limiting) Distribution

As $n \to \infty$, the $n$-step transition probabilities converge to a steady state distribution $\pi$, such that:

$$\pi_j = \lim_{n \to \infty} P^n(x)$$

The steady state distribution satisfies: $$\pi P = \pi \quad \text{and} \quad \sum_{j} \pi_j = 1$$

6. Irreducibility

A Markov chain is irreducible if every state can be reached from every other state, i.e., all states communicate with each other.

7. Recurrence and Transience

• A state $i$ is recurrent if the process returns to state $i$ with probability 1.
• A state $i$ is transient if there is a positive probability of never returning to state $i$.

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Summary Table