Question

Mean and standard deviation of senior citizens in the 2010 US Census. We learn that 13% of all people in the US are senior citizens. Let the random variable X represent the number of senior citizens in a random sample of 10 people. Find the mean and standard deviation of this random variable.

1. Mean (μ) of X:
2. Standard Deviation (σ) of X:

Choose the correct values from the options below:

a) μ = 1.3, σ = 1.14
b) μ = 1.3, σ = 1.07
c) μ = 1.7, σ = 1.14
d) μ = 1.7, σ = 1.07

Answer 1

The mean (μ) for the random variable representing the number of senior citizens in a sample of 10 people is 1.3, and the standard deviation (σ) is approximately 1.07.

Step-by-step explanation:

The question asks us to find the mean (μ) and standard deviation (σ) of the random variable X, which represents the number of senior citizens in a random sample of 10 people from a population where 13% are senior citizens. To find these statistics, we assume a binomial distribution because each person in the sample can be classified as a senior or not.

The mean μ of X is found using the formula μ = np, where n is the sample size and p is the probability of success (in this case, being a senior citizen).

Mean: μ = n * p = 10 * 0.13 = 1.3

The standard deviation σ of X is found using the formula σ = √(np(1-p)), which includes the probability of failure (1-p).

Standard Deviation: σ = √(10 * 0.13 * (1-0.13)) σ = √(10 * 0.13 * 0.87) σ = √(1.131) σ ≈ 1.07

Therefore, the correct option is b) μ = 1.3, σ = 1.07.