Distribusi Normal part 1 - Eksplorasi Soal Distribusi Normal

Normal Distribution Problem Solving (Finding Mean and Standard Deviation)
📌 The first problem requires finding the mean ($\mu$) and standard deviation ($\sigma$) given two Z-scores ($Z_1 = -1.5, Z_2 = 1.5$) and their corresponding raw scores ($X_1 = 65, X_2 = 95$).
📐 The calculation involves setting up and solving a system of two linear equations derived from the Z-score formula: $Z = \frac{X - \mu}{\sigma}$.
✅ The resulting mean was $\mu = 80$ and the standard deviation was $\sigma = 10$.

Calculating Probability in Normal Distribution (Case 1: Upper Bound)
📌 A company produces ink with a mean ($\mu$) of 500 sheets and a standard deviation ($\sigma$) of 50 sheets. The goal is to find the probability that the ink can print not more than 650 sheets, $P(X \le 650)$.
⚙️ The raw score $X=650$ is converted to a Z-score: $Z = \frac{650 - 500}{50} = 3$.
📊 The probability $P(X < 650)$ becomes $P(Z < 3)$, calculated as $P(Z < 0) + P(0 < Z < 3) = 0.5 + 0.4987$.
💡 The final probability calculated is 0.9987. (Note: The speaker emphasizes that if the question asked for a percentage, the final answer must be multiplied by 100, which was not done for the probability result itself in this specific step).

Calculating Percentage in Normal Distribution (Case 2: Lower Bound)
📌 Electrical resistance devices have a mean ($\mu$) of 40 Ohms and a standard deviation ($\sigma$) of 2 Ohms. The task is to find the percentage of devices with resistance exceeding 43 Ohms, $P(X > 43)$.
🛠️ Convert $X=43$ to a Z-score: $Z = \frac{43 - 40}{2} = 1.5$.
📉 To find $P(Z > 1.5)$, it is calculated as $0.5 - P(0 < Z < 1.5)$.
📊 Using the table value $P(0 < Z < 1.5) = 0.4332$, the probability is $0.5 - 0.4332 = 0.0668$.
❗ The final answer required is the percentage, so $0.0668 \times 100 =$6.68%.

Probability Between Two Values (Case 3: Bounded Range)
📌 Given a mean lifespan of 24 months ($\mu=24$) and standard deviation of 6.5 months ($\sigma=6.5$), find the probability a mouse lives between 15 and 33 months, $P(15 < X < 33)$.
🔁 Convert both raw scores to Z-scores: $Z_1 = \frac{15 - 24}{6.5} \approx -1.38$ and $Z_2 = \frac{33 - 24}{6.5} \approx 1.38$.
🧮 Due to symmetry, $P(-1.38 < Z < 1.38)$ is calculated as $2 \times P(0 < Z < 1.38)$.
✅ Using the table value $P(0 < Z < 1.38) = 0.41462$, the final probability is $2 \times 0.41462 =$0.82924.

Finding Raw Score from Percentile (Case 4: Lower Tail)
📌 For trigonometry test scores ($\mu=6.7, \sigma=1.2$), find the highest score achieved by the lowest 10% of students, $P(X < x) = 0.10$.
🔄 The area $P(Z < z) = 0.10$ corresponds to $P(0 < Z < z) = 0.5 - 0.10 = 0.40$.
🔍 Looking up the area $0.40$ in the Z-table yields $Z \approx -1.28$ (since $0.3997$ corresponds to $Z=1.28$).
➕ Convert $Z=-1.28$ back to $X$: $X = \mu + Z\sigma = 6.7 + (-1.28) \times 1.2 =$5.16.

Finding Raw Score from Percentile (Case 5: Upper Tail)
📌 Using the same distribution ($\mu=6.7, \sigma=1.2$), determine the lowest score among the top 30% of students, $P(X > x) = 0.30$.
🔄 The area to the right of the boundary Z-score is $0.30$, meaning the area between the mean (0) and Z is $0.5 - 0.30 = 0.20$.
🔍 Looking up the area $0.20$ in the Z-table suggests a value close to $Z=0.52$ ($P(0 < Z < 0.52) = 0.1985$). Thus, $Z \approx 0.52$.
➕ Convert $Z=0.52$ back to $X$: $X = \mu + Z\sigma = 6.7 + (0.52) \times 1.2 =$7.324. (The speaker calculated $Z=0.52$ and got $X=7.32$).

Key Points & Insights
➡️ Always translate story problems into the proper symbols ($\mu, \sigma, X, Z$) before calculation.
➡️ Use the standard formula $Z = \frac{X - \mu}{\sigma}$ for conversion, and remember $\mu$ is the mean and $\sigma$ is the standard deviation.
➡️ When working with probabilities $P(X < x)$ or $P(X > x)$, utilize the property that the total area under the curve is 1, often splitting the calculation at $Z=0$ (where the area to the left/right is 0.5).
➡️ When finding a raw score ($X$) from a given probability, first find the corresponding Z-score using the table, then convert it back using the rearranged formula: $X = \mu + Z\sigma$.

📸 Video summarized with SummaryTube.com on Feb 02, 2026, 00:40 UTC