Question

In an article in Accounting and Business Research, Carslaw and Kaplan investigate factors that influence “audit delay” for firms in New Zealand. Audit delay, which is defined to be the length of time (in days) from a company’s financial year-end to the date of the auditor’s report, has been found to affect the market reaction to the report. This is because late reports often seem to be associated with lower returns and early reports often seem to be associated with higher returns. Carslaw and Kaplan investigated audit delay for two kinds of public companies-owner controlled and manager-controlled companies. Here a company is considered to be owner controlled if 30 percent or more of the common stock is controlled by a single outside investor (an investor not part of the management group or board of directors). Otherwise, a company is considered manager controlled. It was felt that the type of control influences audit delay. To quote Carslaw and Kaplan: Large external investors, having an acute need for timely information, may be expected to pressure the company and auditor to start and to complete the audit as rapidly as practicable.

(a) Suppose that a random sample of 105 public owner-controlled companies in New Zealand is found to give a mean audit delay of x ¯ x¯ = 82.30 days, and assume that σ equals 31 days. Calculate a 95 percent confidence interval for the population mean audit delay for all public owner-controlled companies in New Zealand. (Round your answers to 3 decimal places.) The 95 percent confidence interval is [ , ]

(b) Suppose that a random sample of 105 public manager-controlled companies in New Zealand is found to give a mean audit delay of x ¯ x¯ = 92 days, and assume that σ equals 39 days. Calculate a 95 percent confidence interval for the population mean audit delay for all public manager-controlled companies in New Zealand. (Round your answers to 3 decimal places.) The 95 percent confidence interval is [ , ] (c) Use the confidence intervals you computed in parts a and b to compare the mean audit delay for all public owner-controlled companies versus that of all public manager-controlled companies. How do the means compare? Mean audit delay for public owner-controlled companies appears to be and there is amount of overlap of the intervals.

Answer 1

a.
TRADITIONAL METHOD
given that,
standard deviation, σ =31
sample mean, x =82.3
population size (n)=105
I.
standard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
standard error = ( 31/ sqrt ( 105) )
= 3.025
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 3.025
= 5.93
III.
CI = x ± margin of error
confidence interval = [ 82.3 ± 5.93 ]
= [ 76.37,88.23 ]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
standard deviation, σ =31
sample mean, x =82.3
population size (n)=105
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 82.3 ± Z a/2 ( 31/ Sqrt ( 105) ) ]
= [ 82.3 - 1.96 * (3.025) , 82.3 + 1.96 * (3.025) ]
= [ 76.37,88.23 ]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 95% sure that the interval [76.37 , 88.23 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean

b.
TRADITIONAL METHOD
given that,
standard deviation, σ =39
sample mean, x =92
population size (n)=105
I.
standard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
standard error = ( 39/ sqrt ( 105) )
= 3.806
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 3.806
= 7.46
III.
CI = x ± margin of error
confidence interval = [ 92 ± 7.46 ]
= [ 84.54,99.46 ]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
standard deviation, σ =39
sample mean, x =92
population size (n)=105
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 92 ± Z a/2 ( 39/ Sqrt ( 105) ) ]
= [ 92 - 1.96 * (3.806) , 92 + 1.96 * (3.806) ]
= [ 84.54,99.46 ]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 95% sure that the interval [84.54 , 99.46 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean

Answers:
a.
95% sure that the interval [76.37 , 88.23 ]
b.
95% sure that the interval [84.54 , 99.46 ]
c.
parts a and b to compare the mean audit delay for all public
owner-controlled companies versus that of all public manager-controlled companies.
Mean audit delay for public owner-controlled companies appears to be and
there is little amount of overlap of the intervals.