A company sells seeds and claims that 55% of its pea seeds germinate - Edexcel - A-Level Maths Statistics - Question 5 - 2017 - Paper 2

Let S = number of seeds out of 24 that germinate, where S follows the binomial distribution, S ~ B(24, 0.55). The number of trays is T = 10. We want to calculate P(T ≥ 5).

Using the binomial probability formula:

$P(T ext{ successes}) = {n ext{ choose } k} p^{k} (1-p)^{n-k}$

Where k is the number of successes (trays with at least 15 seeds germinating) and n is the total number of trials (trays). Calculating further shows that:

$P(T ext{ successes}) ≈ 0.149$

1. The number of trials n is large.
2. The probability of success p is close to 0.5.

Let X ~ N(132, 59.4) where the mean μ = np = 240 * 0.55 and the standard deviation σ = √(np(1-p)). To find P(X ≥ 149.5), we convert to the z-score:

Z = rac{X - ext{mean}}{ ext{std. dev}} = rac{149.5 - 132}{59.4}

Calculating this gives:

$P(X ext{ successes}) ≈ 0.01158.$

Therefore, the approximated probability is about 0.0116.

The probability is very small; therefore, there is evidence that the company's claim is incorrect, suggesting that the actual proportion of germinating seeds may differ from 55%.