Question

A random sample of 100 observations produced a sample proportion of 0.25. An approximate 95% confidence interval for the population proportion p is between?

a. 0.165 and 0.335
b. 0.165 and 0.321
c. 0.179 and 0.321
d. 0.207 and 0.293
e. 0.152 and 0.348

Answer 1

The confidence interval for the population proportion p is between ( 0.165 and 0.335 )

Hence, Option a) 0.165 and 0.335 is the correct answer.

Explanation:

Given the data in the question;

sample size n = 100

sample proportion p₀ = 0.25

q₀ = 1 - p₀ = 1 - 0.25 = 0.75

confidence interval = 95%

First we calculate our Standard Error

Standard Error = √( ( p₀ × q₀ ) / n )

we substitute

Standard Error = √( ( 0.25 × 0.75 ) / 100 )

Standard Error = √( 0.1875 / 100 )

Standard Error = √0.001875

Standard Error = 0.0433

Now, For 95% confidence interval; z = 1.96

so confidence interval for the population proportion p will be;

⇒ p₀ ± ( z × Standard Error )

we substitute

⇒ 0.25 ± ( 1.96 × 0.0433 )

⇒ 0.25 ± 0.084868

⇒ ( 0.25 - 0.084868 ), (0.25 + 0.084868 )

⇒ ( 0.25 - 0.084868 ), (0.25 + 0.084868 )

⇒ ( 0.165, 0.335 )

Therefore, The confidence interval for the population proportion p is between ( 0.165 and 0.335 )

Hence, Option a) 0.165 and 0.335 is the correct answer.