In: Statistics and Probability
In examining the credit accounts of a department store, an auditor would like to estimate the true mean account error (book value – audited value). To do so, the auditor selected a random sample of 50 accounts and found the mean account error to be $60 with a standard deviation of $30.
a. Construct a 95% confidence interval for the population mean account error. What conclusion can be made from this confidence interval?
b. How large a sample is actually needed to ensure with 95% confidence in estimating the population mean account error to within ±$5?
c. Among the 50 selected accounts, the auditor found that there were 6 of them consisted of an account error of more than $100. Based on this information, construct a 99% confidence interval for the population proportion of accounts with an error of more than $100. What conclusion can be made from this confidence interval?
Answer:
a)
Given,
To determine the 95% confidence interval
Sample size n = 50
xbar = 60
standard deviation = s = 30
degrees of freedom = n - 1
= 50 - 1
df = 49
alpha = 1 - 0.95 = 0.05
alpha/2 = 0.05/2 = 0.025
crotical value t (0.025;59) = 2.00958
Now for the 95% confidence interval for the population mean account error.
i.e.,
(xbar - t*s/sqrt(n) , xbar+t*s/sqrt(n) )
substitute values
= ( 60 - 2.00958 *30 /sqrt(50) , 60 - 2.00958 * 30 / sqrt(50))
= (51.474 , 68.526)
lower limit = 51.474
upper limit = 68.526
b)
To determine the sample size
z critical value for 95% confidence interval = 1.96
E = margin of error = 5
standard deviation = sigma = 30
n = (z*sigma/E)^2
now substitute values
= (1.96*30/5)^2
=138.2976
=138 (approximately)
sample size = 138
c)
To determine the 99% confidence interval
sample proportion = p^ = x/n
= 6/50
p^ = 0.12
z critical value for 99% = 2.576
99% confidence interval = (p^-z*sqrt(p^(1-p^)/n , p^+z*sqrt(p^(1-p^)/n )
= (0.12 - 2.576*sqrt(0.12*(1-.12)/50 , 0.12 + 2.576*sqrt(0.12*(1-.12)/50))
= (0.00162 , 0.23838)
So interval limit is 0.00162 < p < 0.23838
lower limit = 0.00162
upper limit = 0.23838
So the interval lies between 0.00162 and 0.23838