Question

1. A botanist surveys a forest for disease and finds that 15 of the 230 white pines sampled have blister rust. Find a 99% confidence interval on the true proportion of white pines with blister rust within the forest. Describe with an English sentence that uses the lower and upper bounds of the confidence interval mean.

2. Suppose the botanist wants to estimate the true proportion of white pines with blister rust to within 1% of the sampled proportion. Still at 99% confidence, how many trees must be sampled to determine this?

Answer 1

Solution :

Given that,

Point estimate = sample proportion = = x / n = 15/230 = 0.0652

1 - = 0.9348

1)  

Z/2 = 2.58

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 2.58 * [(0.0652 * 0.9348) / 230]

= 0.0420

A 99% confidence interval for population proportion p is ,

- E < p < + E

0.0652 - 0.0420 < p < 0.0652 + 0.0420

0.0232 < p < 0.1072

Lower bound = 0.0232

Upper bound = 0.1072

2)

margin of error = E = 0.01

sample size = n = (Z / 2 / E)2 * * (1 - )

= (2.58 / 0.01)2 * (0.0652 * 0.9348)

= 4057

4,057 trees must be sampled