Question 1191863: Your mother said, if in the 3rd Quarter your grade in Math is 95 and above, she will add Php50 to your daily allowance, Php40 if 90 - 94, Php30 if 85 - 89, Php20 if 80-84, but will decrease your allowance by Php15 if your grade is 79 and below. If the probability of getting 95 and above is 17%, 90 - 94 is 15%, 85 - 89 is 28%, 80 - 84 is 25%, and 15% if 79 and below. Construct the probability distribution of the additional allowances your mother will give you. Find the expected value, variance, and standard deviation.(use two decimal place for the mean, variance and standard deviation)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's the solution:
**1. Define the Random Variable:**
Let X be the random variable representing the additional allowance your mother will give you.
**2. Construct the Probability Distribution:**
| Grade Range | Additional Allowance (X) | Probability (P(X)) |
|---|---|---|
| 95 and above | Php 50 | 0.17 |
| 90 - 94 | Php 40 | 0.15 |
| 85 - 89 | Php 30 | 0.28 |
| 80 - 84 | Php 20 | 0.25 |
| 79 and below | -Php 15 | 0.15 |
**3. Calculate the Expected Value (Mean):**
The expected value E(X) is calculated as the sum of each possible value of X multiplied by its probability:
E(X) = (50 * 0.17) + (40 * 0.15) + (30 * 0.28) + (20 * 0.25) + (-15 * 0.15)
E(X) = 8.5 + 6 + 8.4 + 5 - 2.25
E(X) = 25.65
Therefore, the expected additional allowance is Php 25.65.
**4. Calculate the Variance:**
The variance Var(X) is calculated as the expected value of the squared difference between each value of X and the mean:
Var(X) = Σ[(X - E(X))² * P(X)]
Var(X) = (50 - 25.65)² * 0.17 + (40 - 25.65)² * 0.15 + (30 - 25.65)² * 0.28 + (20 - 25.65)² * 0.25 + (-15 - 25.65)² * 0.15
Var(X) = (24.35)² * 0.17 + (14.35)² * 0.15 + (4.35)² * 0.28 + (-5.65)² * 0.25 + (-40.65)² * 0.15
Var(X) = 593.9225 * 0.17 + 205.9225 * 0.15 + 18.9225 * 0.28 + 31.9225 * 0.25 + 1652.4225 * 0.15
Var(X) = 100.97 + 30.89 + 5.30 + 7.98 + 247.86
Var(X) = 392.99
Therefore, the variance of the additional allowance is 392.99.
**5. Calculate the Standard Deviation:**
The standard deviation SD(X) is the square root of the variance:
SD(X) = √Var(X)
SD(X) = √392.99
SD(X) ≈ 19.82
Therefore, the standard deviation of the additional allowance is approximately Php 19.82.
**Summary of Results:**
* **Expected Value (Mean):** Php 25.65
* **Variance:** 392.99
* **Standard Deviation:** Php 19.82
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