In: Statistics and Probability
According to Thomson Financial, last year the majority of companies reporting profits had beaten estimates. A sample of 162 companies showed that 100 beat estimates, 29 matched estimates, and 33 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*= |
Solution :
Given that,
n = 162
x = 33
a) Point estimate = sample proportion = = x / n = 33 / 162 = 0.2037
1 - = 1 - 0.2037 = 0.7963
b) At 95% confidence level
= 1 - 95%
=1 - 0.95 =0.05
/2
= 0.025
Z/2
= Z0.025 = 1.960
Margin of error = E = Z / 2 * (( * (1 - )) / n)
ME = 1.96 (((0.2037 * 0.7963) / 162 )
ME = 0.0620
c) margin of error = E = 0.05
sample size = n = (Z / 2 / E )2 * * (1 - )
= (1.96 / 0.05)2 * 0.2037 * 0.7963
= 249.25
sample size = n = 250