Questions: American households increasingly rely on cell phones as their exclusive telephone service. It is reported that 50.2% of American households still have landline phone service. We decide to randomly call eight households and ask if the home has a landline phone. Required: a-1. What is the random variable? The random variable is the count of the households that have a landline. a-2. How is the random variable distributed? The random variable follows a probability distribution.

American households increasingly rely on cell phones as their exclusive telephone service. It is reported that 50.2% of American households still have landline phone service. We decide to randomly call eight households and ask if the home has a landline phone.

Required:
a-1. What is the random variable?

The random variable is the count of the households that have a landline.
a-2. How is the random variable distributed?

The random variable follows a probability distribution.

Transcript text: American households increasingly rely on cell phones as their exclusive telephone service. It is reported that $50.2 \%$ of American households still have landline phone service. We decide to randomly call eight households and ask if the home has a landline phone. Required: a-1. What is the random variable? The random variable is the count of the households that have a landline. a-2. How is the random variable distributed? The random variable follows a probability distribution.

Solution

Solution Steps

Step 1: Define the Random Variable

The random variable $ X $ represents the count of households that have a landline phone among the 8 randomly called households.

Step 2: Determine the Distribution

The random variable $ X $ follows a binomial distribution, denoted as $ X \sim \text{Binomial}(n=8, p=0.502) $, where:

$ n = 8 $ is the number of trials (households called),
$ p = 0.502 $ is the probability that a household has a landline phone.

Step 3: Calculate Probabilities

Using the binomial probability formula $ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} $, where $ q = 1 - p = 0.498 $, we find the probabilities for $ x = 0, 1, 2 $:

For $ x = 0 $: \[ P(X = 0) = \binom{8}{0} \cdot (0.502)^0 \cdot (0.498)^8 = 0.0038 \]
For $ x = 1 $: \[ P(X = 1) = \binom{8}{1} \cdot (0.502)^1 \cdot (0.498)^7 = 0.0305 \]
For $ x = 2 $: \[ P(X = 2) = \binom{8}{2} \cdot (0.502)^2 \cdot (0.498)^6 = 0.1076 \]

Step 4: Calculate Statistical Measures

The mean $ \mu $, variance $ \sigma^2 $, and standard deviation $ \sigma $ of the binomial distribution are calculated as follows:

Mean: \[ \mu = n \cdot p = 8 \cdot 0.502 = 4.016 \]
Variance: \[ \sigma^2 = n \cdot p \cdot q = 8 \cdot 0.502 \cdot 0.498 = 2.0 \]
Standard Deviation: \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{8 \cdot 0.502 \cdot 0.498} = 1.4142 \]