Question

a teacher asks her students to write down the number of hours studied, rounded to the nearest half hour. she compiles the results and develops the probability distribution for a randomly selected student. what is the mean of the probability distribution? probability distribution hours studied: x probability: p(x) 0.5 0.07 1 0.2 1.5 0.46 2 0.2 2.5 0.07 0.85 1.50 0.46 0.20

Explanation:

Step1: Recall the formula for the mean of a probability distribution

The mean (expected value) \( \mu \) of a discrete probability distribution is calculated as \( \mu = \sum (X \cdot P(X)) \), where \( X \) is the value of the random variable and \( P(X) \) is its corresponding probability.

Step2: Calculate each \( X \cdot P(X) \) term

• For \( X = 0.5 \) and \( P(X) = 0.07 \): \( 0.5 \times 0.07 = 0.035 \)
• For \( X = 1 \) and \( P(X) = 0.2 \): \( 1 \times 0.2 = 0.2 \)
• For \( X = 1.5 \) and \( P(X) = 0.46 \): \( 1.5 \times 0.46 = 0.69 \)
• For \( X = 2 \) and \( P(X) = 0.2 \): \( 2 \times 0.2 = 0.4 \)
• For \( X = 2.5 \) and \( P(X) = 0.07 \): \( 2.5 \times 0.07 = 0.175 \)

Step3: Sum up all the \( X \cdot P(X) \) terms

\( 0.035 + 0.2 + 0.69 + 0.4 + 0.175 \)
First, add \( 0.035 + 0.2 = 0.235 \)
Then, \( 0.235 + 0.69 = 0.925 \)
Next, \( 0.925 + 0.4 = 1.325 \)
Finally, \( 1.325 + 0.175 = 1.5 \)

Answer:

1.50