Question

1) There is an average of 2.1 weedy species per square metre in a native grassland. What is the probability that a square metre has at most 1 weedy species?

2)

For the next poll of the university’s president's approval rating, we want to be within 1% of the true population proportion and use a 95% confidence level. How many individuals should we sample? In the last poll his approval rate was 72%, so we will use that as our sample proportion.

		78
		3317
		7745
		13378

Answer 1

1)

Here, λ = 2.1 and x = 1
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X <= 1).
P(X <= 1) = (2.1^0 * e^-2.1/0!) + (2.1^1 * e^-2.1/1!)
P(X <= 1) = 0.1225 + 0.2572
P(X <= 1) = 0.3797

2)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.01

The provided estimate of proportion p is, p = 0.72
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.72*(1 - 0.72)*(1.96/0.01)^2
n = 7744.67

Therefore, the sample size needed to satisfy the condition n >= 7744.67 and it must be an integer number, we conclude that the minimum required sample size is n = 7745
Ans : Sample size, n = 7745