Forty percent of participants in a soccer league are nine years old, 30% are eight, 20% are seven, and 10% are
six. What is the expected value of the age and the standard deviation?
To find the expected value and standard deviation of the ages in the soccer league, we need to calculate the weighted average of the ages based on the given percentages.
Step 1: Assign values to each age group:
- Nine years old: 9
- Eight years old: 8
- Seven years old: 7
- Six years old: 6
Step 2: Calculate the expected value:
The expected value, also known as the mean, is the sum of the products of each value and its corresponding probability.
Expected Value (μ) = (9 * 0.40) + (8 * 0.30) + (7 * 0.20) + (6 * 0.10)
Step 3: Calculate the standard deviation:
Standard deviation (σ) is a measure of how spread out the data is. We can use the following formula to calculate it:
Standard Deviation (σ) = √[(∑(x - μ)^2 * P(x)]
where:
- x represents each value
- μ represents the expected value
- P(x) represents the probability of each value
For simplicity, let's use a table to calculate the standard deviation:
Age | Probability (P) | (x - μ) | (x - μ)^2 | (x - μ)^2 * P
---------------------------------------------------------
9 | 0.40 | 0.6 | 0.36 | 0.144
---------------------------------------------------------
8 | 0.30 | 0.4 | 0.16 | 0.048
---------------------------------------------------------
7 | 0.20 | 0.3 | 0.09 | 0.018
---------------------------------------------------------
6 | 0.10 | 1.3 | 1.69 | 0.169
---------------------------------------------------------
Sum of (∑(x - μ)^2 * P(x)): 0.379
Standard Deviation (σ) = √(0.379)
Therefore, the expected value of the ages in the soccer league is 7.8 (rounded to one decimal place), and the standard deviation is approximately 0.616 (rounded to three decimal places).