Question

A youth director for a city's parks and recreations department is planning for next year's program. From a survey of city residents, the director knows that 35% of the population participate in the parks and recreations programs. For a random sample of 40 people, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places?

Answer 1

Given Information:  

Sample size = n = 40  

sample proportion = p = 35% = 0.35  

Required Information:

standard deviation of sample proportion = σ

Answer:  

standard deviation of sample proportion = σ = 0.075

Step-by-step explanation:  

Check if the condition np ≥  10 or n(1 - p) ≥ 10 is satisfied

np ≥ 10

40*0.35 ≥ 10

14 ≥ 10 satisfied

n(1 - p) ≥ 10

40(1 - 0.35) ≥ 10

26 ≥ 10 satisfied

Which means that distribution of sample proportion will have a mean closer to the population proportion.

The standard deviation for sample proportion is given by

$\sigma = \sqrt{\frac{p(1-p)}{n}}$

$\sigma = \sqrt{\frac{0.35(1-0.35)}{40}}$

$\sigma = 0.075$

Therefore, the standard deviation for the sampling distribution of the sample proportions is 0.075.