It is known that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?

Question

BlackTom AI · Accepted Answer

The question asks about a binomial scenario where 40% of voters favor a candidate and a sample of size 30 is observed. The variable of interest is the number of supporters in the sample, call it X.

First, restating the key setup: X ~ Binomial(n = 30, p = 0.4). The condition “less than 35% are in favour” translates to fewer than 0.35 × 30 = 10.5 supporters. Since the number of supporters must be an integer, this is equivalent to X ≤ 10.

Next, outline the calculation approach. There are two common routes:
- Exact binomial calculation: P(X ≤ 10) = sum from k=0 to 10 of C(30, k) (0.4)^k (0.6)^(30−k).
- Approximate approach (often used for quick checks): use a normal approximation with continuity correction. For X ~ Binomial(30, 0.4), the mean is μ = np = 12 and the variance is σ^2 = np(1−p) = 30 × 0.4 × 0.6 = 7.2, so σ ≈ 2.683.
  Applying continuity correction for P(X ≤ 10) gives P(X ≤ 10.5). Compute z = (10.5 − μ) / σ ≈ (10.5 − 12) / 2.683 ≈ −0.56. The standard normal CDF Φ(z) at z ≈ −0.56 is about 0.28 to 0.29, roughly 0.288.

Why this result makes sense: the mean 12 is a bit higher than 10, so P(X ≤ 10) is a left tail probability. The approximate value around 0.288 aligns with the idea that getting 10 or fewer successes in this setup is less than a third of the time.

Important note about the answer options: the provided data does not list any answer choices. As a result, there is no explicit set of lettered options to evaluate directly. If you were given options such as (a) 0.15, (b) 0.20, (c) 0.288, (d) 0.50, you could map the approximate binomial calculation to the closest value and then justify why the others are too large or too small given the distribution Around the mean. Without the actual options, the key takeaway is understanding how to translate the percentage threshold (35% or 10.5 voters) into a binomial tail probability and how the normal approximation yields a value near 0.288 for this scenario.

It is known that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?单项选择题

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