Question

Cercarial dermatitis, commonly known as "swimmer's itch", is a rash caused by an allergic reaction to parasites that burrow into the skin. The parasites are commonly found in waterfowl such as the Canadian goose. A group of researchers examined a random sample of 142 Canadian geese and found that 59 were infected with the parasite. Use this data to find the lower limit of a 98% confidence interval for the true proportion of Canadian geese that are affected by the parasite. Give your answer rounded to 4 decimal places.

Answer 1

Solution :

n = 142

x = 59

$\hat p$ = x / n = 59 / 142 = 0.415

1 -$\hat p$ = 1 - 0.415 = 0.585

At 98% confidence level the z is ,

$\alpha$ = 1 - 98% = 1 - 0.98 = 0.02

$\alpha$ / 2 = 0.02 / 2 = 0.01

Z_$\alpha$/2 = Z_0.01 = 2.326

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$(($\hat p$ * (1 - $\hat p$)) / n)

= 2.326 * ($\sqrt$((0.415 * 0.585) / 142)

= 0.096

A 98 % confidence interval for population proportion p is ,

$\hat p$- E < P <$\hat p$ + E

0.415 - 0.096 < p < 0.415 + 0.096

0.319 < p < 0.511