Comprehensive Study Guide: Statistics, Probability, and Complex Numbers

Measures of Central Tendency and Data Spread

Median Calculation: The median is defined as the average of the two middle values after the data set has been sorted in ascending or descending order.
Specific Problem (Q47): * Dataset provided: $6, 68, 93, 82, 76, 110$. (Based on scores of 6 students). * Sorting the data: $6, 68, 76, 82, 93, 110$. * The middle two values are $76$ and $82$. * Calculation: $\text{Median} = \frac{76 + 82}{2} = 79$ .
Measures of Dispersion: * These include the Range, Variance, and Standard Deviation. * The Arithmetic Mean is a measure of central tendency, not dispersion.
Comparing Standard Deviation (Q49): * To find the data set with the largest standard deviation, look for the set with the greatest spread (Range). * A useful technique is to delete identical values among choices to identify which range is largest. * Data Sets: * A: $14, 10, 12, 11, 13, 13$ * B: $14, 10, 15, 11, 13, 13$ * C: $11, 10, 20, 11, 13, 13$ * D: $14, 10, 30, 11, 13, 13$ * Dataset D has the largest outlier ( $30$ ), resulting in the largest range and thus the largest standard deviation.

Normal Distribution Analysis

Empirical Rule Application (Q50): * Total population: $10,000$ batteries. * Mean ( $\mu$ ): $300$ days. * Standard Deviation ( $\sigma$ ): $40$ days. * Range of interest: $260$ to $340$ days. * This range represents $\mu - \sigma$ to $\mu + \sigma$ . * According to the normal distribution percentage, $68\%$ of data falls within one standard deviation. * Calculation: $0.68 \times 10,000 = 6,800$ batteries.
Probability and Percentages (Q51-Q53): * Scenario 1: $\mu = 2$ , $\sigma = 1$ . To find the probability that x > 3: * $3 = \mu + 1\sigma$ . * The percentage above $1\sigma$ is calculated by subtracting the values below it ( $50\% + 34\% = 84\%$ ) from $100\%$ . * Result: $16\%$ . * Scenario 2: $\mu = 25$ , $\sigma = 2$ . To find the probability that x < 27: * $27 = \mu + 1\sigma$ . * The area to the left of $+1\sigma$ includes the mean and below ( $50\%$ ) plus the region between the mean and $+1\sigma$ ( $34\%$ ). * Result: $84\%$ . * Scenario 3: $\mu = 12$ , $\sigma = 2$ . To find P(10 < x < 16). * Lower bound $10 = \mu - 1\sigma$ ; Upper bound $16 = \mu + 2\sigma$ . * Area breakdown: $\text{Between } \mu-1\sigma \text{ and } \mu = 34\%$ , $\text{Between } \mu \text{ and } \mu+1\sigma = 34\%$ , $\text{Between } \mu+1\sigma \text{ and } \mu+2\sigma = 13.5\%$ . * Total probability: $34\% + 34\% + 13.5\% = 81.5\%$ .
Finding Parameters from Range (Q54-Q55): * Standard Deviation from 99% Range: If $99\%$ of students score between $13$ and $49$ . * In prep materials, $99\%$ is often treated as covering approximately $6\sigma$ (from $\mu - 3\sigma$ to $\mu + 3\sigma$ ). * $49 - 13 = 36$ . * $6\sigma = 36 \implies \sigma = 6$ . * Note: The excluded $1\%$ represents $0.5\%$ on the right tail and $0.5\%$ on the left tail. * Mean from 95% Range: If $95\%$ of students weigh between $57$ and $65$ . * The mean is the midpoint of the range. * Calculation: $\mu = \frac{57 + 65}{2} = 61$ .

Binomial Distribution and Skewness

Visual Analysis of Skewness: * A graph is described as having a Positive Skew (Skewed Right) when the tail extends toward the positive direction (higher values) on the x-axis.
Binomial Mean and Standard Deviation Formulas: * Mean: $\mu = n \times p$ * Variance: $\sigma^2 = n \times p \times q$ * Standard Deviation: $\sigma = \sqrt{n \times p \times q}$ * Where $n$ is the number of trials, $p$ is the probability of success, and $q$ is the probability of failure ( $1-p$ ).
Calculating the Mean (Q57): * Given: $p = 35\%$ (or $0.35$ ), $n = 4$ . * Calculation: $\mu = 4 \times 0.35 = 1.4$ .
Calculating Standard Deviation (Q58): * Given: $n = 20$ , $\mu = 12$ . * First, find $p$ : $12 = 20 \times p \implies p = 0.6$ . * Then, find $q$ : $q = 1 - 0.6 = 0.4$ . * Calculate Standard Deviation: $\sigma = \sqrt{20 \times 0.6 \times 0.4} = \sqrt{12 \times 0.4} = \sqrt{4.8}$ .

Complex Numbers and De Moivre's Theorem

De Moivre's Theorem Formula: For a natural number $n$ , $z^n = [r(\cos(\theta) + i \sin(\theta))]^n = r^n(\cos(n\theta) + i \sin(n\theta))$ .
Power Calculation Examples: * Example 1: $[2(\cos(22.5^\circ) + i \sin(22.5^\circ))]^4$ * $2^4 = 16$ * Angle: $4 \times 22.5^\circ = 90^\circ$ * Result: $16(\cos(90^\circ) + i \sin(90^\circ)) = 16(0 + i \times 1) = 16i$ . * Example 2: $(\cos(15^\circ) + i \cos(75^\circ))^6$ * Using the identity $\cos(\theta) = \sin(90^\circ - \theta)$ , we find $\cos(75^\circ) = \sin(15^\circ)$ . * The expression simplifies to $(\cos(15^\circ) + i \sin(15^\circ))^6$ . * Applying De Moivre's: $\cos(6 \times 15^\circ) + i \sin(6 \times 15^\circ) = \cos(90^\circ) + i \sin(90^\circ) = i$ .
Division of Complex Numbers: $\frac{r_1(\cos(\theta_1) + i \sin(\theta_1))}{r_2(\cos(\theta_2) + i \sin(\theta_2))} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$ . * Example: $\frac{20(\cos(80^\circ) + i \sin(80^\circ))}{4(\cos(20^\circ) + i \sin(20^\circ))} = 5(\cos(60^\circ) + i \sin(60^\circ))$ * Rectangular conversion: $5(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = 2.5 + 2.5\sqrt{3}i$ .

Roots of Complex Numbers

n-th Root Formula: $W_k = \sqrt[n]{r} [\cos(\frac{\theta + 2k\pi}{n}) + i \sin(\frac{\theta + 2k\pi}{n})]$ for $k = 0, 1, 2, …, n-1$ .
Key Properties: * The Modulus of all roots is the same: $\sqrt[n]{r}$ . * Example (Q18/Q20): Finding the third roots of $8(\cos(\theta) + i \sin(\theta))$ . The modulus of any root (including the second root) is $\sqrt[3]{8} = 2$ . * Roots of Unity: The modulus of any $n$-th root of $1$ is always $1$ .
Finding the Argument (Amplitude) of Roots: * Example: Finding the fifth roots of $\sqrt{3}(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$ . * The argument of the first root ( $k=0$ ) is $\frac{\theta}{n} = \frac{\pi/3}{5} = \frac{\pi}{15}$ .

Polar Coordinates and Equations

Coordinate Representation: A point is given by $(r, \theta)$ , where $r$ is the directed distance from the pole (origin) and $\theta$ is the directed angle from the polar axis.
Equivalent Representations: * $(r, \theta)$ is equivalent to $(r, \theta + 360^\circ k)$ or $(-r, \theta + 180^\circ)$ .
Distance in Polar Coordinates: The distance between $(r_1, \theta_1)$ and $(r_2, \theta_2)$ is determined by the formula: $d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_2 - \theta_1)}$ . * Example: Distance between $(5, 120^\circ)$ and $(2, 300^\circ)$ . The angle difference is $180^\circ$ . Distance is $|5| + |2| = 7$ . (Calculation: $\sqrt{5^2 + 2^2 - 2(5)(2)\cos(180^\circ)} = \sqrt{25 + 4 - 20(-1)} = \sqrt{49} = 7$ ).
Polar Curves Identification: * $r = a$ : Represents a circle centered at the pole with radius $|a|$ . * $\theta = \alpha$ : Represents a straight line passing through the pole with a slope of $\tan(\alpha)$ .
Conversions (Polar to Rectangular): * $x = r \cos(\theta)$ , $y = r \sin(\theta)$ . * Example Equation: $y = x^2$ transformed to polar form. * $r \sin(\theta) = (r \cos(\theta))^2$ * $r \sin(\theta) = r^2 \cos^2(\theta)$ * $r = \frac{\sin(\theta)}{\cos^2(\theta)} = \tan(\theta) \sec(\theta)$ .

Vocabulary and Principles

Probability Distribution: A function that connects each outcome of the sample space to its probability of occurring.
Correlation: A relationship between two events or variables.
Bias: A design in a survey or study that intentionally favors certain results over others.
Control Group: A group in an experimental study that receives a formal treatment (or no treatment) that has no actual effect on the result, used for comparison.
Margin of Error: Defines the interval representing the potential difference in response between the sample and the population.
Argand Plane: A coordinate system where the x-axis represents the real parts and the y-axis represents the imaginary parts of complex numbers.
Modulus: The absolute value or distance from the origin for a complex number.
Amplitude (Symmetry): The angle $\theta$ in the polar form of a complex number.
Pole: The fixed point (origin) in the polar coordinate system.
Polar Axis: The horizontal ray extending to the right from the pole.