Question

A psychic was tested for ESP. The psychic was presented with 400 cards face down and was asked to determine if the card was one of four symbols: a cross, a star, a circle, or a square. The psychic was correct in 120 of the cases. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial.

A. Using the results above, construct a 99% confidence interval for p.

a	(.255, .345)
b	(.262, .338)
c	(.241, .359)
d	(.259, .341)
e	We can assert that p = 0.25 with 100% confidence because the psychic is just guessing.

B. How large a sample n would you need to estimate p with a margin of error of 0.01 with 95% confidence? (Use the guess 0.25 as the value of p)

a	N = 447
b	N = 7203
c	N = 9604
d	N = 30

Answer 1

Solution :

Given that,

A)

Point estimate = sample proportion = = x / n = 120 / 400 = 0.300

Z_/2 = 2.576

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

= 2.576 * (((0.300 * 0.700) / 400)

= 0.059

A 99% confidence interval for population proportion p is ,

- E < p < + E

0.300 - 0.059 < p < 0.300 + 0.059

0.241 < p < 0.359

(0.241 , 0.359)

B)

= 0.25

1 - = 0.75

margin of error = E = 0.01

Z_/2 = 1.96

sample size = n = (Z _{/ 2} / E)² * * (1 - )

= (1.96 / 0.01)² * 0.25* 0.75

= 7203

sample size = n = 7203