Question

A random sample of n = 500 observations from a binomial population produced x = 250 successes. Find a 90% confidence interval for p. (Round your answers to three decimal places.)

Interpret the interval:

A. In repeated sampling, 90% of all intervals constructed in this manner will enclose the population proportion.

B. 90% of all values will fall within the interval.

C. In repeated sampling, 10% of all intervals constructed in this manner will enclose the population proportion.

D. There is a 10% chance that an individual sample proportion will fall within the interval.

E. There is a 90% chance that an individual sample proportion will fall within the interval.

Answer 1

Solution :

Given that,

n = 500

x = 250

Point estimate = sample proportion = = x / n = 0.5

1 - = 0.5

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z _0.05 = 1.645

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 * (((0.5*0.5) / 500)

= 0.037

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.5 - 0.037 < p < 0.5 + 0.037

0.463 < p < 0.537

The 90% confidence interval for the population proportion p is : ( 0.463 , 0.537 )

E)

There is a 90% chance that an individual sample proportion will fall within the interval.