Question

In: Statistics and Probability

A company’s manager is considering simplification of a travel voucher form. In order to assess the...

A company’s manager is considering simplification of a travel voucher form. In order to assess the costs associated with erroneous travel vouchers, the manager must estimate the total number of such vouchers that were filled out incorrectly in the last month. In a random sample of 100 vouchers drawn without replacement from the 1,440 travel vouchers submitted in the last month, 33 vouchers were filled out incorrectly.(a) Find a point estimate of and a 99 percent confidence interval for the true proportion of travel vouchers that were filled out incorrectly in the last month. (b) Find a point estimate of and a 99 percent confidence interval for the total number of travel vouchers that were filled out incorrectly in the last month. (c-1) If it costs the company $11 to correct an erroneous travel voucher, find a reasonable estimate of the minimum cost of correcting all of last month’s erroneous travel vouchers.

Expert Solution

Solution:

(a) Find a point estimate of and a 99 percent confidence interval for the true proportion of travel vouchers that were filled out incorrectly in the last month. We are given

Sample size = n = 100

x = number of items of interest = 33

Confidence interval for Population Proportion is given as below:

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

Point estimate for true proportion = P = x/n = 33/100 = 0.33

Confidence level = 99%

Critical Z value = 2.5758

(by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.33 ± 2.5758* sqrt(0.33*(1 – 0.33)/n)

Confidence Interval = 0.33 ± 2.5758* 0.0470

Confidence Interval = 0.33 ± 0.1211

Lower limit = 0.33 - 0.1211 = 0.2089

Upper limit = 0.33 + 0.1211 = 0.4511

Confidence interval = (0.2089, 0.4511)

(b) Find a point estimate of and a 99 percent confidence interval for the total number of travel vouchers that were filled out incorrectly in the last month.

We have

Point estimate for true proportion = P = x/n = 33/100 = 0.33

Population size = N = 1440

Point estimate for the total number of travel vouchers that were filled out incorrectly in the last month is given as below:

Required point estimate = N*P = 1440*0.33 = 475.2

99 percent confidence interval for the total number of travel vouchers that were filled out incorrectly in the last month is given as below:

Confidence interval = (1440*0.2089, 1440*0.4511)

Confidence interval = (300.816, 649.584)

(c-1) If it costs the company $11 to correct an erroneous travel voucher, find a reasonable estimate of the minimum cost of correcting all of last month’s erroneous travel vouchers.

Lower limit for confidence interval for the total number of travel vouchers that were filled out incorrectly in the last month is given as 300.816.

Cost per erroneous travel voucher = $11

Minimum cost of correcting all of last month’s erroneous travel vouchers = $11*300.816

Required minimum cost = $3308.976

Required minimum cost = $3309 approximately

orchestra answered 2 years ago

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