Question 1199322: The height of a compartment is normally distributed with a mean of 205 and a deviation of 2 inches.
Inside this compartment we put boxes, the height of these boxes has a mean of 50 and a deviation of 2 inches.
a) What function on the graphing calculator do you need to use?
b) What is the probability that we wont be able to stack 4 boxes in this compartment.
(The correct answer is: 0.132)
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a) Function on the Graphing Calculator**
* **Normal Cumulative Distribution Function (CDF):** You'll primarily use the normal CDF function on your calculator. This function calculates the probability that a random variable from a normal distribution falls below a certain value.
**b) Probability of Not Being Able to Stack 4 Boxes**
1. **Determine the Required Compartment Height:**
* To stack 4 boxes, the compartment height must be greater than or equal to the sum of the heights of the four boxes.
* Let X1, X2, X3, and X4 represent the heights of the four boxes.
* Required Compartment Height ≥ X1 + X2 + X3 + X4
2. **Find the Mean and Standard Deviation of the Total Box Height:**
* **Mean:** Since the box heights are independent and normally distributed, the mean of the sum of their heights is the sum of their individual means:
* Mean of Total Box Height = 50 + 50 + 50 + 50 = 200 inches
* **Variance:** The variance of the sum of independent random variables is the sum of their variances:
* Variance of Total Box Height = 2² + 2² + 2² + 2² = 16
* Standard Deviation of Total Box Height = √16 = 4 inches
3. **Calculate the Probability of Insufficient Compartment Height:**
* We need to find the probability that the sum of the box heights exceeds the compartment height.
* Let C represent the compartment height.
* We want to find P(C < X1 + X2 + X3 + X4)
* **Using the calculator's normal CDF function:**
* Input the following:
* Lower Bound: -∞ (or a very large negative number)
* Upper Bound: 205 (compartment height)
* Mean: 200 (mean of total box height)
* Standard Deviation: 4 (standard deviation of total box height)
* The calculator will give you the probability that the total box height exceeds the compartment height, indicating the probability that we won't be able to stack 4 boxes.
**In Summary**
* Use the normal CDF function on your calculator to find the probability that the sum of four normally distributed box heights exceeds the compartment height.
* Remember to use the mean and standard deviation of the total box height in the calculation.
**Note:** This approach assumes that the heights of the boxes are independent of each other.
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