Question

The population proportion is 0.26. What is the probability that a sample proportion will be within ±0.04 of the population proportion for each of the following sample sizes? (Round your answers to 4 decimal places.)

(a)

n = 100

(b)

n = 200

(c)

n = 500

(d)

n = 1,000

(e)

What is the advantage of a larger sample size?

There is a higher probability

σ_p

will be within ±0.04 of the population standard deviation.We can guarantee

p

will be within ±0.04 of the population proportion p.    As sample size increases,

E(p)

approaches p.There is a higher probability

p

will be within ±0.04 of the population proportion p.

Answer 1

We would be looking at the first 4 parts here as:

a) For a sample size of n = 100, the distribution of the proportion is given as:

The probability here is computed as:
= P( 0.22 < p < 0.3)

Converting it to a standard normal variable, we get:

Getting it from the standard normal tables, we get here:

Therefore 0.638 is the required probability here.

b) For a sample size of n = 200, the probability is computed here as:

Therefore, the probability here is computed as:

Getting it from the standard normal tables, we get here:

therefore 0.802 is the required probability here.

c) For a sample size of 500,

Therefore, the probability now is computed here as:

Getting it from the standard normal tables, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.958 is the required probability here.

d) For a sample size of 1000, we have here:

Therefore the probability now is computed here as:

Getting it from the standard normal tables, we get here:

Therefore 0.996 is the required probability here.