A recent report found that 25% of U.S. households still have a landline telephone. Suppose a random sample of 200 homes was taken and a resident of the home was asked, “Do you have a landline telephone in your place of residence?” Of those asked, 46 said that they have a landline telephone.
What is the probability that in the sample no more than 46 have a landline phone?

This is a problem of probability, specifically involving a **binomial distribution** because we're dealing with a fixed number of trials (200 homes), two possible outcomes (households either have a landline or don't), and a constant probability of success (25% or 0.25).

We are asked to find the probability that **no more than 46 households** have a landline. This means we need to calculate the cumulative probability for \( X \leq 46 \), where \( X \) is the number of households with a landline in a sample of 200 homes.

### Step 1: Define the parameters of the binomial distribution
- \( n = 200 \) (the number of trials, or the number of homes sampled),
- \( p = 0.25 \) (the probability of success, i.e., the probability a household has a landline),
- \( X \) is the number of households with a landline in the sample of 200 homes.

The random variable \( X \) follows a **binomial distribution**, i.e., \( X \sim \text{Binomial}(n = 200, p = 0.25) \).

### Step 2: Approximate using the normal distribution
Since \( n \) is large, we can use the **normal approximation to the binomial distribution**. The conditions for using the normal approximation are met because both \( np \geq 5 \) and \( n(1-p) \geq 5 \).

- Mean \( \mu = np = 200 \times 0.25 = 50 \),
- Standard deviation \( \sigma = \sqrt{np(1-p)} = \sqrt{200 \times 0.25 \times 0.75} = \sqrt{37.5} \approx 6.12 \).

### Step 3: Apply continuity correction
To improve the approximation, we use a continuity correction because we're approximating a discrete distribution (binomial) with a continuous one (normal).

We want \( P(X \leq 46) \), so we approximate this by finding \( P(X \leq 46.5) \) using the normal distribution.

### Step 4: Standardize the value
We standardize 46.5 to a z-score:

\[
z = \frac{46.5 - 50}{6.12} \approx \frac{-3.5}{6.12} \approx -0.57
\]

### Step 5: Find the probability
Now, using the standard normal distribution, we find the probability associated with \( z = -0.57 \).

Using a standard normal table or a calculator, the cumulative probability corresponding to \( z = -0.57 \) is approximately 0.2843.

Thus, the probability that no more than 46 households in the sample have a landline is approximately **0.2843**, or **28.43%**.

---

Would you like further details on this calculation or another example? Here are 5 related questions to expand this topic:

1. How do you calculate binomial probabilities directly without using the normal approximation?
2. What is the exact probability that exactly 46 households have a landline?
3. What if the probability of having a landline changes to 30%? How does this affect the result?
4. How do you interpret z-scores and their probabilities in the context of normal distributions?
5. Can the normal approximation be used for any binomial distribution?

**Tip**: For large sample sizes, the normal approximation is a quick and effective method for binomial probabilities!

A recent report found that 25% of U.S. households still have a landline telephone. Suppose a random sample of 200 homes was taken and a resident of the home was asked, 'Do you have a landline telephone in your place of residence?' Of those asked, 46 said that they have a landline telephone. What is the probability that in the sample no more than 46 have a landline phone?

This problem explores the probability that no more than 46 out of 200 sampled households have a landline phone, using a binomial distribution with normal approximation. The problem involves key concepts such as mean, standard deviation, Z-scores, and cumulative probability. Suitable for advanced high school and college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation