Algebra 2 Semester 2 Final Review: Sequences, Probability, and Statistics Notes and Stats

Arithmetic and Geometric Sequences

Sequence Problem 1

• Sequence Table:
  • $n=0$, $f(n)=4.2$
  • $n=1$, $f(n)=6.3$
  • $n=2$, $f(n)=9.45$
  • $n=3$, $f(n)=14.175$
• Type of Sequence: Geometric
• Ratio Calculation: $6.3 \div 4.2 = 1.5$
• Initial Term ($a_1$ as per transcript): $a_1 = 4.2$
• Common Difference/Ratio: $1.5$
• Explicit Rule: $a_n = 4.2(1.5)^{n-1}$
• Recursive Rule: $a_1 = 4.2$, $a_n = a_{n-1} \times 1.5$

Sequence Problem 2

• Sequence Table:
  • $n=0$, $f(n)=17.3$
  • $n=1$, $f(n)=13.5$
  • $n=2$, $f(n)=9.7$
  • $n=3$, $f(n)=5.9$
• Difference Calculation: $13.5 - 17.3 = -3.8$
• Type of Sequence: Arithmetic
• Initial Term: $17.3$
• Common Difference/Ratio: $-3.8$
• Explicit Rule: $a_n = 17.3 - 3.8n$
• Recursive Rule: $a_1 = 17.3$, $a_n = a_{n-1} - 3.8$

Sequence Problem 3

• Sequence Table:
  • $n=0$, $f(n)=\frac{1}{2}$
  • $n=1$, $f(n)=2$
  • $n=2$, $f(n)=8$
  • $n=3$, $f(n)=32$
• Type of Sequence: Geometric
• Initial Term: $a_0 = \frac{1}{2}$
• Common Difference/Ratio: $\times 4$
• Explicit Rule: $a_n = \frac{1}{2}(4)^n$
• Recursive Rule: $a_1 = \frac{1}{2}$, $a_n = a_{n-1} \times 4$

Sequence Problem 4

• Sequence Table:
  • $n=1$, $f(n)=-24$
  • $n=2$, $f(n)=12$
  • $n=3$, $f(n)=-6$
  • $n=4$, $f(n)=3$
  • $n=5$, $f(n)=-1.5$
• Type of Sequence: Geometric
• Initial Term: $a_1 = -24$
• Common Difference/Ratio: $\times -0.5$
• Explicit Rule: $a_n = -24(-0.5)^{n-1}$
• Recursive Rule: $a_1 = -24$, $a_n = a_{n-1} \times -0.5$

Arithmetic Sequence Reconstruction (Problem 5)

• Given: $a_5 = 14$ and $a_9 = 54$. Initial term index is $0$.
• Difference Calculation: $\frac{54-14}{9-5} = \frac{40}{4} = 10$. Common Difference ($d$) is $10$.
• Initial Term ($a_0$) Calculation:
  • $a_5 = 14$
  • $a_4 = 4$
  • $a_3 = -6$
  • $a_2 = -16$
  • $a_1 = -26$
  • $a_0 = -36$
• Recursive Rule: $a_0 = -36$, $a_n = a_{n-1} + 10$
• Explicit Rule: $a_n = -36 + 10n$

Geometric Sequence Reconstruction (Problem 6)

• Given: $a_2 = 36$ and $a_5 = 324$. Initial term index is $0$.
• Ratio Calculation:
  • $36 \times r^3 = 324$
  • $r^3 = \frac{324}{36} = 9$ (Note: Transcript calculation steps show $\sqrt[3]{r^3} = \sqrt[3]{9}$ leading to $r=3$; note $3^3 = 27$, however calculation written is $r^3=9 \rightarrow r=3$ as interpreted in solving $\sqrt{9}=3$).
• Initial Term ($a_0$) Calculation:
  • $a_2 = 36$
  • $a_1 = 12$
  • $a_0 = 4$
• Explicit Rule: $a_n = 4(3)^n$
• Recursive Rule: $a_0 = 4$, $a_n = a_{n-1} \times 3$

Probability of Compound Events

Dice Experiments

1. Rolling two fair dice and adding the dots (Sum is 7 or 11):

   • Sum 7 possibilities: $\frac{6}{36}$
   • Sum 11 possibilities: $\frac{2}{36}$
   • Total probability: $\frac{8}{36} = \frac{2}{9} \approx 0.22$ ($22\%$

2. Rolling two fair dice and adding the dots (Sum is even or a perfect square):

   • Total probability: $\frac{19}{36} \approx 0.53$ ($53\%$

Standard Deck of Cards

3. Drawing a card (Red or an Ace):

   • $P(\text{Red}) = \frac{26}{52}$
   • $P(\text{Ace}) = \frac{4}{52}$
   • $P(\text{Red and Ace}) = \frac{2}{52}$
   • Calculation: $\frac{26}{52} + \frac{4}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13} \approx 0.54$ ($54\%$

4. Drawing a card (Black or Even):

   • $P(\text{Black}) = \frac{26}{52}$
   • Calculation recorded: $\frac{26}{52} + \frac{30}{52} - \frac{10}{52} = \frac{46}{52} = \frac{23}{26} \approx 0.88$ ($88\%$

5. Drawing 2 cards without replacement (Face card then a 10):

   • Calculation: $\frac{12}{52} \times \frac{4}{51} = \frac{48}{2652} = \frac{4}{221} \approx 0.018$ ($1.8\%$

6. Drawing 2 cards without replacement (Queen then another Queen):

   • Calculation: $\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221} \approx 0.0045$ ($0.45\%$

Selection from a Box (With Replacement)

7. Box with 6 red, 7 blue, and 2 green pens (Blue then Green):

   • Total pens = $15$
   • Calculation: $\frac{7}{15} \times \frac{2}{15} = \frac{14}{225} \approx 0.062$ ($6.2\%$

8. Box with 6 red, 4 blue, and 4 green pens (Red or Blue then Green):

   • Total pens = $14$
   • Calculation: $\frac{10}{14} \times \frac{4}{14} = \frac{5}{7} \times \frac{2}{7} = \frac{10}{49} \approx 0.20$ ($20\%$

Two-Way Tables and Marginal Frequencies

Exercise 3: Test Preparation and Outcomes

• a. Probability that a person studied, failed the test: $\frac{4}{38} \approx 0.105$ ($10.5\%$
• b. Probability that a person passed the test: $\frac{40}{50} = \frac{4}{5} = 0.8$ ($80\%$

Exercise 4: Survey Responses by Role

• a. Probability that a teacher responded yes: $\frac{3}{10} = 0.30$ ($30\%$
• b. Probability that a person surveyed is a student: $\frac{98}{108} = \frac{49}{54} \approx 0.91$ ($91\%$

Exercise 5: Hand Washing Survey Results

• Marginal Frequencies Interpretation: Total males = $171$, Total females = $180$, Total surveyed = $351$. Total wash = $283$, Total no wash = $68$.
• a. Probability person does not wash hands: $\frac{68}{351} \approx 0.19$ ($19\%$
• b. Probability that a female washes her hands: $\frac{151}{180} \approx 0.84$ ($84\%$

Unit Circle, Radians, and Trig Identities

Unit Circle Coordinates

• Angle $150^\circ$: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$ (Quadrant II).
• Angle $315^\circ$: $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$ (Quadrant IV).

Unit Conversions

• Convert $45^\circ$ to radians: $45 \times \frac{\pi}{180} = \frac{\pi}{4}$.
• Convert $225^\circ$ to radians: $225 \times \frac{\pi}{180} = \frac{5\pi}{4}$.
• Convert $\frac{3\pi}{4}$ to degrees: $\frac{3 \times 180}{4} = 135^\circ$.
• Convert $\frac{2\pi}{3}$ to degrees: $\frac{2 \times 180}{3} = 120^\circ$.

Trig Identities

• $\cos^2(\theta) + \sin^2(\theta) = 1$
• $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
• $\cos^2(\theta) = 1 - \sin^2(\theta)$
• $\sin^2(\theta) = 1 - \cos^2(\theta)$

Data Collection and Standard Deviation

Data Collection Terms

• Population: The entire group being studied (e.g., $250$ computers, $100$ movie viewers).
• Sample: The subset of the population actually tested/surveyed (e.g., 25 randomly chosen computers, every 30th person).\n* **Sampling Method:**\n * **Simple Random:** Testing randomly chosen items.\n * **Systematic:** Surveying every n$-th person (e.g., every$30-th exiting viewer).\n\n## Standard Deviation Definition\n* The standard deviation is the average distance from the **mean**.\n* The standard deviation is the **square root** of the variance.\n\n## Dataset Analysis\n\n### Dataset 1: 4, 5, 8, 1, 2, 3, 9, 8, 7, 6, 2\n* **Mean:** 5\n* **Median:** 5\n* **Standard Deviation:** 2.66\n* **Range:** 8\n* **Interpretation:** The numbers in the list are an average of 2.66 units away from the mean.\n\n### Dataset 2: 23, 10, 11, 24, 32, 15, 13, 16\n* **Mean:** 18\n* **Median:** 15.5\n* **Standard Deviation:** 7.14\n* **Range:** 22\n* **Interpretation:** The average distance from the mean is 7.14.\n\n# Box-and-Whisker Plots\n\n## Plot Reading: Number of Minutes to Read a Chapter\n* **Data points:** 10, 22, 25, 35, 45\n* **Lower Extreme (Min):** 10\n* **Upper Extreme (Max):** 45\n* **Median (Q_2$):**$25\n* **Lower Quartile (Q_1$):**$22\n* **Upper Quartile (Q_3$):**$35\n* **Range between Lower and Upper Quartile:** Q_3 - Q_1 = 35 - 22 = 13\n* **Range where data is most clustered:** Between Q_1$and$Q_2$(Width =$3)\n* **Range where data is most spread out:** Between Q_1$and$Q_3$(Width =$13)\n* **Interquartile Range (IQR):** 35 - 22 = 13$$
• Distribution Type: Skewed Right