Question

According to Thomson Financial, last year the majority of companies reporting profits had beaten estimates. A sample of 162 companies showed that 96 beat estimates, 29 matched estimates, and 37 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.
	n*=

Answer 1

1. What is the point estimate of the proportion that fell short of estimates

P -short = x/n

Where,

x is number of fell short = 37

n is the sample size = 162

Therefore

P -short = 37/162 = 0.2284

2. Determine the margin of error and provide a 95% confidence interval for the proportion that beat estimates.

We have,

Mean x = 96

Sample size n = 162

Confidence interval = 95%

Therefore z = 1.96 at 95% confidence interval

And p = x/n = 96/162 = 0.5926

Therefore,

The margin of error (ME) = z *√ (p *(1-p)/n)

= 1.96 *√ (0.5926 *(1-0.5926)/162) = 0.0757

The boundaries of Confidence interval = p – ME and p + ME

= 0.5926 - 0.0757 and 0.5926 + 0.0757= 0.5169 and 0.6683

3. How large a sample is needed if the desired margin of error is 0.05?

n = p *(1-p) (z/e) ^2

Where,

Sample size, n =?

p = 0.5926

z = 1.96

Desired margin of error e = 0.05

Therefore,

n = (0.5926)*(1 -0.5926) (1.96/.05)^2 = 370.99 or 371

Therefore sample size should be 371.