Question

A grower believes that one in five of his citrus trees are infected with the citrus red mite. How large a sample should be taken if the grower wishes to estimate the proportion of his trees that are infected with citrus red mite to within 0.08 with probability 0.9?

Answer 1

Answer: 68

Step-by-step explanation:

Formula for sample size when prior estimate of population proportion (p) is available:  $n=p(1-p)(\dfrac{z^*}{E})^2$

, where z*= critical-value.

E= Margin of error.

Let p be the population proportion of  trees are infected with the citrus red mite.

As per given , we have

$p=\dfrac{1}{5}=0.2$

E= ± 0.08

The critical z-value corresponding to 90% confidence level = z*=1.645

Substitute all the values in the above formula , we get

Required sample size  :$n=(0.2)(1-0.2)(\dfrac{(1.645)}{0.08})^2$

$\Rightarrow\ n=(0.2)(0.8)(20.5625)^2$

$\Rightarrow\ n=0.16(422.81640625)\\\\\Rightarrow\ n=67.650625\approx68$ [Rounded to next integer.]

Thus, the minimum sample size should be taken =68