BAN 530
Inference Examples
1. A new car dealer has determined that the dealership must average more than 4.8% profit on the sales of its new cars to allow an increase in the commission paid his sales representatives. A random sample of 80 recently sold new cars yielded a mean percentage profit of 4.87% with a standard deviation of 3.9%. Does the data suggest the average percentage profit is sufficient to increase the commission? Use a 0.01 level of significance.
Ho: μ = 4.8
Ha: μ > 4.8
0.16
Reject Ho if t 2.375
Fail to Reject Ho
There is insufficient evidence to conclude the average percentage profit is sufficient to increase the commission, α = 0.01.
2. In July 1995, it was reported that the average net income for sole practitioner CPAs was about $75,000 a year. The Texas State Board of Public Accountancy believes that sole practitioner CPAs in Texas have a lower average net income than this figure. To test this claim, the Board took a random sample of 112 sole practitioner CPAs in Texas and found the mean yearly net income to be $72,220. Assume the standard deviation of net incomes for sole practitioner CPAs in Texas is $14,530. At the 0.03 level of significance, does the data support the Board’s claim?
Ho: μ = 75,000
Ha: μ 75,000
-2.025
Reject Ho if Z -1.89
Reject Ho
There is sufficient evidence to support the Board’s claim, α = 0.03.
3. A large manufacturing company investigated the service it has received from suppliers and discovered that, in the past, 32% of all material shipments have been received late. The company installed a just-in-time system of ordering materials to address this concern. A random sample of 118 deliveries since the new system was installed revealed 22 deliveries were late. At the 0.02 level of significance, has the proportion of late deliveries been reduced?
Ho: π = .32
Ha: π .32
-3.120
Reject Ho if Z -2.06
Reject Ho
There is sufficient evidence to conclude the proportion of late deliveries been reduced, α = 0.02.
4. Health insurers and the federal government are both putting pressure on hospitals to shorten the average length of stay (LOS) of their patients. A random sample of 20 hospitals in one state had a mean LOS in 1990 of 3.8 days and a standard deviation of 1.2 days. Use a 90% confidence interval to estimate the true mean LOS for the state’s hospitals in 1990 and interpret.
90% C.I. for μ:
3.8 ± 0.464
(3.336, 4.264)
We are 90% confident that the true mean LOS for the state’s hospitals in 1990 was between 3.336 and 4.264 days.
5. Shoplifting is an escalating problem for retailers. Recently, one New York City store randomly selected 500 shoppers and observed them while they were in the store. Out of the 500 shoppers, 40 were seen stealing. Construct a 94% confidence interval for the true proportion of all the store’s customers who are shoplifters and interpret.
94% C.I. for π:
0.08 ± 0.023
(0.057, 0.103)
We are 94% confident the true proportion of all the store’s customers who are shoplifters is between 0.057 and 0.103.
6. It is believed that the IQ scores of fourth graders are normally distributed with a population standard deviation of 15 IQ points. A random sample of 100 fourth graders is obtained and a mean IQ score of 110 is found. Construct a 97% confidence interval for the true mean IQ scores of fourth graders and interpret.
97% C.I. for μ:
110 ± 3.255
(106.745, 113.255)
We are 97% confident the true mean IQ scores of fourth graders is between 106.745 and 113.255.