Question

The population proportion of success is 8% and the intended sample size is n=824. Before drawing a sample from the population, you first want to estimate the 85-th percentile, that is the value separating the lower 85% of the sample proportions from the upper 15% of the sample proportions.

For this problem, the normal approximation will be used. First, it is necessary to calculate the “critical value”, which is the z-score separating the lower 85% of the standard normal distribution from the upper 15%:
       zc.v.=

Percentile for the sample proportions:
       P85=
Give answers in decimal format (as opposed to fractions or percentages).

Answer 1

Solution

Given that,

p = 0.08

1 - p = 1 - 0.08 = 0.92

n = 824

= p = 0.08

=  [p ( 1 - p ) / n] =   [(0.08 * 0.92) / 824 ] = 0.0095

Using standard normal table,

P(Z < z) = 85%

= P(Z < z) = 0.85  

= P(Z < 1.036) = 0.85

z = 1.036

Using z-score formula,

  = z *   +  

  = 1.036 * 0.0095 + 0.08

  = 0.089

  = 8.9%