To solve this, we need to calculate the probability using the binomial distribution, as the problem involves binary outcomes (a household either has or does not have a landline).

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Given Data:

• Probability of success ($p$): $0.25$
• Sample size ($n$): $200$
• Number of successes ($x$): $46$

We are looking for $P(X \leq 46)$, which is the cumulative probability up to $x = 46$.

Solution Steps:

1. Check if the problem can be approximated by a normal distribution:

The binomial distribution can be approximated by the normal distribution if $n \cdot p \geq 5$ and $n \cdot (1 - p) \geq 5$.

• $n \cdot p = 200 \cdot 0.25 = 50$
• $n \cdot (1 - p) = 200 \cdot 0.75 = 150$

Both conditions are satisfied, so we can use the normal approximation.

2. Parameters of the normal distribution:

For a binomial distribution approximated by a normal distribution:

• Mean ($\mu$) = $n \cdot p = 50$
• Standard deviation ($\sigma$) = $\sqrt{n \cdot p \cdot (1 - p)}$ $\sigma = \sqrt{200 \cdot 0.25 \cdot 0.75} = \sqrt{37.5} \approx 6.123$

3. Convert $X$ to a Z-score:

Using the continuity correction, we calculate the Z-score for $x = 46.5$ (adjusted for continuity correction): $Z = \frac{x - \mu}{\sigma} = \frac{46.5 - 50}{6.123} \approx \frac{-3.5}{6.123} \approx -0.572$

4. Find the cumulative probability:

Using the Z-score table, $P(Z \leq -0.572)$: $P(Z \leq -0.572) \approx 0.2831$

Thus, $P(X \leq 46) = 0.2831$.

The correct answer is:

**None of the options match exactly (likely due to rounding inconsistencies). Let me cross-check.from scipy.stats import binom, norm

Given parameters

n = 200 p = 0.25 x = 46

Mean and standard deviation for normal approximation

mu = n * p sigma = (n * p * (1 - p))**0.5

Z-score with continuity correction

z = (x + 0.5 - mu) / sigma

Cumulative probability using normal approximation

cumulative_prob = norm.cdf(z)

cumulative_prob0.283814249631667960.28381424963166796successThe cumulative probability $P(X \leq 46)$, calculated using the normal approximation, is approximately 0.2838. This does not match the options provided in the question.

The discrepancy suggests either a rounding error or an issue with the options' precision. Let me confirm the exact binomial calculation to ensure the correct result.# Exact cumulative probability using binomial distribution exact_cumulative_prob = binom.cdf(x, n, p)

exact_cumulative_prob0.28700472608957990.2870047260895799successThe exact cumulative probability $P(X \leq 46)$, calculated using the binomial distribution, is approximately 0.2870.

This result still does not match the provided options. It suggests that the question may have errors in the listed choices or that the exact computation was not expected. Based on either approximation or exact calculation, the closest value is 0.2870.

Would you like further clarification or to explore more on this topic?

5 Questions for Contextual Understanding:

1. What are the conditions for using a normal approximation to a binomial distribution?
2. How does a continuity correction improve the accuracy of normal approximation?
3. What tools are most efficient for solving binomial problems in practice?
4. How can you interpret the significance of the cumulative probability in this context?
5. Why might the given answer choices differ from precise computed probabilities?

Tip:

For higher accuracy, always prefer the exact binomial computation unless the dataset is large enough to justify an approximation.