Question

suppose that an accounting firm does a study to determine the time needed to complete one persons tax forms. it randomly surveys 100 people. the sample mean is 23.6 hours. there is a known standard deviation of 7.0 hours. the population distribution is assumed to be normal.

a) if the firm wished to increase it level of confidence and keep the error bound the same by taking another survey, what changes should it make?

b) if the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? why?

c) suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. how would the number of peopke the firm surveys change? why?

Answer 1

Part (a)

Population standard deviation = 7 hours

According to the error bound formula,

where sigma is the population standard deviation, n is the sample size and z is the critical value for 96% confidence interval.

When we increase the confidence level, the z value also increases. Also, the sample standard deviation is constant. Thus, from the error bound formula, in order to keep EBM constant and increase the confidence level, we must increase the sample size.

Part (b)

The confidence level will decrease.

According to the error bound formula,

where sigma is the population standard deviation, n is the sample size and z is the critical value for 96% confidence interval.

Since EBM and sigma are constant. If we decrease the sample size from 100 to 49, the z - value will decrease, and hence the confidence level will also decrease.

Part (c)

Population standard deviation = 7 hours

Error bound margin, EBM = 1 hour

Confidence level required = 96%

According to the error bound formula:

where sigma is the population standard deviation, n is the sample size and z is the critical value for 96% confidence interval.

Now,

From the table,

Now, if n= 100,

Now, since we have decreased the error bound margin from 1.435 hours to 1 hour, from the error bound formula the sample size should increase. This is verified below:

From the error bound formula we get,

Thus,

Thus, n=206 people are required for the survey.