Phase 1: The Core Concepts

To solve this problem, you need to understand Maximum Likelihood Estimation (MLE).

1. What is a Likelihood Function ($L$)?

Imagine you have a die, but you don’t know if it’s fair. You roll it a few times and get specific numbers. The Likelihood is the probability of getting that specific sequence of numbers, assuming a specific variable (parameter) is true.

• Formula: $L(\theta) = P(x_1) \cdot P(x_2) \cdot P(x_3) \dots$ (We multiply the probabilities because each roll is an “independent event” — one roll doesn’t affect the next).

2. What is Maximum Likelihood Estimation (MLE)?

In statistics, we work backward. We have the data (the rolls), and we want to find the “best guess” for the probability ($\theta$). The MLE is simply the value of $\theta$ that makes the observed data most likely to have happened. To find this “maximum” point, we use calculus (derivatives).

3. Why use Log-Likelihood ($\ln L$)?

The Likelihood function usually involves multiplying many small numbers ($x \cdot y \cdot z$). Calculus is hard with multiplication (Product Rule). Calculus is easy with addition. By taking the natural log ($\ln$) of the function, we turn multiplication into addition:

• Log Rule: $\ln(a \cdot b) = \ln(a) + \ln(b)$
• Power Rule: $\ln(x^n) = n \cdot \ln(x)$

This makes the derivative much easier to solve, and the maximum point remains at the same spot.

Phase 2: Step-by-Step Solution

Step 1: Define the Probabilities

From the question text:

• The probability of getting a 1, 2, or 3 is $\theta$.
  • $P(1) = \theta$
  • $P(2) = \theta$
  • $P(3) = \theta$
• The probability of getting a 4 is the remaining probability.
  • $P(4) = 1 - 3\theta$

Step 2: Analyze the Data

The observed outcomes are: 1, 4, 3, 4, 3, 2, 4

Let’s count how many times each type of probability appears:

• Type A (Outcome 1, 2, or 3):
  • We have a 1, a 3, a 3, and a 2.
  • Total count ($n_A$) = 4
• Type B (Outcome 4):
  • We have a 4, a 4, and a 4.
  • Total count ($n_B$) = 3

Total rolls = 7.

Step 3: Build the Likelihood Function

We multiply the probability of each specific roll that occurred. $L(\theta) = \theta \cdot (1-3\theta) \cdot \theta \cdot (1-3\theta) \cdot \theta \cdot \theta \cdot (1-3\theta)$

Group them together using exponents: $L(\theta) = \theta^4 \cdot (1-3\theta)^3$

(Read this as: We got the "$\theta$" outcome 4 times, and the "$1-3\theta$" outcome 3 times).

Step 4: Take the Log-Likelihood

Now, apply the natural log ($\ln$) to make it solvable. $l(\theta) = \ln( L(\theta) )$ $l(\theta) = \ln( \theta^4 \cdot (1-3\theta)^3 )$

Using the Log Rules ($\ln(ab) = \ln a + \ln b$ and $\ln(x^n) = n\ln x$): $l(\theta) = 4\ln(\theta) + 3\ln(1-3\theta)$

Step 5: Differentiate and Maximize

To find the maximum, we take the derivative with respect to $\theta$ and set it to equal 0.

Recall the derivative formulas:

1. $\frac{d}{dx} \ln(x) = \frac{1}{x}$
2. Chain Rule: $\frac{d}{dx} \ln(u) = \frac{1}{u} \cdot u'$

Let’s derive our equation: $l'(\theta) = \frac{d}{d\theta}[4\ln(\theta)] + \frac{d}{d\theta}[3\ln(1-3\theta)]$

1. First part: $4 \cdot (\frac{1}{\theta}) = \frac{4}{\theta}$
2. Second part (Chain Rule applies here because the inside is $1-3\theta$, not just $\theta$):
   • Derivative of outer $\ln$ function: $3 \cdot \frac{1}{1-3\theta}$
   • Multiplied by derivative of inside function ($1-3\theta$): $-3$
   • Result: $3 \cdot \frac{1}{1-3\theta} \cdot (-3) = \frac{-9}{1-3\theta}$

Combine them: $l'(\theta) = \frac{4}{\theta} - \frac{9}{1-3\theta}$

Step 6: Solve for $\theta$

Set the derivative to 0 to find the maximum likelihood estimate.

$\frac{4}{\theta} - \frac{9}{1-3\theta} = 0$

Move the negative term to the right side: $\frac{4}{\theta} = \frac{9}{1-3\theta}$

Cross-multiply: $4(1 - 3\theta) = 9\theta$

Expand: $4 - 12\theta = 9\theta$

Add $12\theta$ to both sides: $4 = 21\theta$

Divide by 21: $\theta = \frac{4}{21}$

Step 7: Final Calculation

Now we convert the fraction to a decimal. $4 \div 21 \approx 0.190476...$

The question asks for the answer correct to two decimal places. $0.1904... \rightarrow \mathbf{0.19}$

Final Answer

The maximum likelihood estimate of $\theta$ is 0.19.

Solution to Question 1: Maximum Likelihood Estimation (MLE)

1. The Concept: What is MLE? Maximum Likelihood Estimation is a method to find the most likely value of a parameter (in this case, $\theta$) that would produce the data you actually observed.

• Likelihood Function ($L$): The probability of seeing your specific data sequence. Since die rolls are independent, we multiply the probabilities of each roll.
• Log-Likelihood ($\ln L$): We take the natural log of $L$ to turn multiplication into addition, making it easier to take the derivative.

2. Step-by-Step Solution

Step A: Define Probabilities

• Probability of rolling 1, 2, or 3: $P(1)=P(2)=P(3) = \theta$
• Probability of rolling 4: $P(4) = 1 - 3\theta$

Step B: Analyze Observed Data Sequence: $\{1, 4, 3, 4, 3, 2, 4\}$

• Count of outcome $\{1, 2, 3\}$: We have a 1, a 3, a 3, and a 2. (Total count = 4)
• Count of outcome $\{4\}$: We have three 4s. (Total count = 3)

Step C: Build the Likelihood Function $L(\theta) = (\theta)^4 \cdot (1 - 3\theta)^3$

Step D: Take the Log-Likelihood ($l$) Using the log rules $\ln(a \cdot b) = \ln(a) + \ln(b)$ and $\ln(x^n) = n \cdot \ln(x)$: $l(\theta) = 4\ln(\theta) + 3\ln(1 - 3\theta)$

Step E: Differentiate and Maximize Take the derivative with respect to $\theta$ and set it to 0. $l'(\theta) = \frac{d}{d\theta}[4\ln(\theta)] + \frac{d}{d\theta}[3\ln(1 - 3\theta)]$ $0 = \frac{4}{\theta} + 3 \cdot \frac{1}{1 - 3\theta} \cdot (-3)$ (Note: The -3 comes from the chain rule on $1-3\theta$) $0 = \frac{4}{\theta} - \frac{9}{1 - 3\theta}$

Step F: Solve for $\theta$ Move the negative term to the other side: $\frac{4}{\theta} = \frac{9}{1 - 3\theta}$

Cross-multiply: $4(1 - 3\theta) = 9\theta$ $4 - 12\theta = 9\theta$ $4 = 21\theta$ $\theta = \frac{4}{21}$

Final Calculation: $\theta \approx 0.19047...$ Rounding to two decimal places: Answer: 0.19

Solution to Question 2: Power of the Test

1. The Concept: Power of a Test The “Power” of a statistical test is the probability that you correctly reject the Null Hypothesis ($H_0$) when the Alternative Hypothesis ($H_A$) is actually true.

• Rejection Region: The set of outcomes where you decide to reject $H_0$.
• Formula: Power $= P(\text{Outcome falls in Rejection Region} \mid \text{True parameter } p)$

2. Step-by-Step Solution

Step A: Analyze the Rejection Condition The problem states we reject $H_0$ if: $|X - 300| > 50$ This inequality splits into two parts:

1. $X - 300 > 50 \implies X > 350$
2. $X - 300 < -50 \implies X < 250$

So, the Rejection Region is $X < 250$ or $X > 350$.

> Teacher’s Note: There is likely a typo in the question numbers ($|X-300| > 50$ for $n=100$? $X$ can only go up to 100). The logic below strictly follows the algebraic form provided in the question options, likely approximating with a much larger scale or using the numbers abstractly.

Step B: Standardize to Z-scores We approximate the Binomial distribution with a Normal distribution.

• Mean ($\mu$): $n \cdot p = 100p$
• Standard Deviation ($\sigma$): $\sqrt{n \cdot p(1-p)} = \sqrt{100p(1-p)} = 10\sqrt{p(1-p)}$

We convert the boundary values ($X = 250$ and $X = 350$) into Z-scores using the formula $Z = \frac{X - \mu}{\sigma}$.

Lower Z-score ($Z_L$) for $X=250$: $Z_L = \frac{250 - 100p}{10\sqrt{p(1-p)}}$ Algebraic Trick: Divide the numerator and denominator by 10 to match the answer options. $Z_L = \frac{25 - 10p}{\sqrt{p(1-p)}}$

Upper Z-score ($Z_U$) for $X=350$: $Z_U = \frac{350 - 100p}{10\sqrt{p(1-p)}}$ Simplify by 10: $Z_U = \frac{35 - 10p}{\sqrt{p(1-p)}}$

Step C: Construct the Probability Function The Power is the probability of being in the rejection region: $\text{Power} = P(Z < Z_L) + P(Z > Z_U)$

Using the Cumulative Distribution Function ($F_Z$ or $\Phi$) representing the area to the left:

• $P(Z < Z_L) = F_Z(Z_L)$
• $P(Z > Z_U) = 1 - F_Z(Z_U)$

Combine them: $\text{Power} = 1 + F_Z(Z_L) - F_Z(Z_U)$

Step D: Match with Options Substitute our simplified $Z_L$ and $Z_U$ back in: $\text{Power} = 1 + F_Z\left(\frac{25 - 10p}{\sqrt{p(1-p)}}\right) - F_Z\left(\frac{35 - 10p}{\sqrt{p(1-p)}}\right)$

Answer: This matches the first option (marked with the green check).