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%.

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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!