Question

According to Thomson Financial, last year the majority of companies reporting profits had beaten estimates. A sample of 162 companies showed that 112 beat estimates, 29 matched estimates, and 21 fell short.

(a)	What is the point estimate of the proportion that fell short of estimates? If required, round your answer to four decimal places.
	pshort=
(b)	Determine the margin of error and provide a 95% confidence interval for the proportion that beat estimates. If required, round your answer to four decimal places.
	ME =
(c)	How large a sample is needed if the desired margin of error is 0.05? If required, round your answer to the next integer.

Answer 1

Solution :

Given that,

n = 162

x = 21

(a)

Point estimate = sample proportion = = x / n = 21 / 162 = 0.130

(b)

x = 112

(a)

Point estimate = sample proportion = = x / n = 112 / 162 = 0.691

1 - = 0.309

Z_/2 = Z _0.025 = 1.96

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

= 1.96 * (((0.691 * 0.309) / 162)

= 0.071

ME = 0.071

(c)

= 0.5

1 - = 1 - 0.5 = 0.5

margin of error = E = 0.05

Z_/2 = Z_0.025 = 1.96

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

= (1.96 / 0.05)² * 0.5 * 0.5

= 385

sample size = n = 385