Question

Alex tracked how many cups of coffee he drank each day for a month and then compiled the results into a probability distribution, as shown below.

Number of Cups of Coffee, upper X 0 1 2 3 4
Relative Frequency, upper P left parenthesis upper X right parenthesis 0.024 0.423 0.291 0.172 0.090
Question
Calculate the IQR of Alex’s probability distribution.

(1 point)
Responses

1.5
1.5

2
2

2.5
2.5

3

Answer 1

To calculate the interquartile range (IQR) of Alex's probability distribution, we first need to determine the cumulative probabilities and identify the first quartile (Q1) and the third quartile (Q3).

### Given Data:
1. **Number of Cups of Coffee (X)**: 0, 1, 2, 3, 4
2. **Relative Frequency (P(X))**: 0.024, 0.423, 0.291, 0.172, 0.090

### Step 1: Calculate Cumulative Distributions
- For 0 cups: \( P(X \leq 0) = 0.024 \)
- For 1 cup: \( P(X \leq 1) = 0.024 + 0.423 = 0.447 \)
- For 2 cups: \( P(X \leq 2) = 0.447 + 0.291 = 0.738 \)
- For 3 cups: \( P(X \leq 3) = 0.738 + 0.172 = 0.910 \)
- For 4 cups: \( P(X \leq 4) = 0.910 + 0.090 = 1.000 \)

### Step 2: Find Q1 and Q3
- **Q1 (25th percentile)**: We look for the smallest \( X \) such that \( P(X \leq X) \geq 0.25 \):
- For \( X = 0\): \( P(X \leq 0) = 0.024 \)
- For \( X = 1\): \( P(X \leq 1) = 0.447 \) (this is the first value greater than 0.25)

Thus, **Q1 = 1**.

- **Q3 (75th percentile)**: We look for the smallest \( X \) such that \( P(X \leq X) \geq 0.75 \):
- For \( X = 2\): \( P(X \leq 2) = 0.738 \)
- For \( X = 3\): \( P(X \leq 3) = 0.910 \) (this is the first value greater than 0.75)

Thus, **Q3 = 3**.

### Step 3: Calculate IQR
\[ \text{IQR} = Q3 - Q1 = 3 - 1 = 2 \]

### Conclusion
The interquartile range (IQR) of Alex's probability distribution is **2**.

Thus, the correct response is:

**2**