Briefly describe how the three quartiles are calculated for a data set. Illustrate by calculating the three quartiles for two examples, the first with an odd number of observations and the second with an even number of observations.
Below is a relative frequency table showing American household incomes as reported in the 2000 Census?,*To simplify your calculations, assume the last income class is $200,000 to $300,000. Notice that the classes become wider at higher levels of income. ,a. Draw the corresponding histogram. ,b. Estimate the mean of household incomes.
(a) n = 50, ˆp = 0.095,(b) n = 100, ˆp= 0.095,(c) n = 500, ˆp = 0.095,(d) n = 1000, ˆp = 0.095,(e) Comment on the effect of sample size on the observed P-value of the test.
(a) Determine the critical value for a right-tailed test of a population mean at the ? = 0.01 level of signi?cance with 15 degrees of freedom.,(b) Determine the critical value for a left-tailed test of a population mean at the ? = 0.05 level of signi?cance based on a sample size of n = 20.,(c) Determine the critical values for a two-tailed test of a population mean at the ? = 0.05 level of signi?cance based on a sample size of n = 13.
(a) At ? = .05, does the following sample show that daughters are taller than their mothers?,(b) Is the decision close?,(c) Why might daughters tend to be taller than their mothers? Why might they not?
(a) A statistical study reported that a drug was effective with a p-value of .042. Explain in words what this tells you.,(b) How would that compare to a drug that had a p-value of .087?
1. To help the manufacturer get a clear picture of type I and type II error probabilities, draw a ? versus ? chart for sample sizes of 30, 40, 60, and 80. If ? is to be at most 1% with ? = 5%, which sample size among these four values is suitable?,2. Calculate the exact sample size required for ? = 5% and ? = 1%. Construct a sensitivity analysis table for the required sample size for ? ranging from 2,788 to 2,794 psi and ? ranging from 1% to 5%.,3. For the current practice of n = 40 and ? = 5% plot the power curve of the test. Can this chart be used to convince the manufacturer about the high probability of passing batches that have a strength of less than 2,800 psi?,4. To present the manufacturer with a comparison of a sample size of 80 versus 40, plot the OC curve for those two sample sizes. Keep an ? of 5%.,5. The manufacturer is hesitant to increase the sample size beyond 40 due to the concomitant increase in testing costs and, more important, due to the increased time required for the tests. The production process needs to wait until the tests are completed, and that means loss of production time. A suggestion is made by the production manager to increase ? to 10% as a means of reducing ?. Give an account of the benefits and the drawbacks of that move. Provide supporting numerical results wherever possible.,When a tire is constructed of more than one ply, the interply shear strength is an important property to check. The specification for a particular type of tire calls for a strength of 2,800 pounds per square inch (psi). The tire manufacturer tests the tires using the null hypothesis where ? is the mean strength of a large batch of tires. ,H0: ? ? 2,800 psi,From past experience, it is known that the population standard deviation is 20 psi.,Testing the shear strength requires a costly destructive test and therefore the sample size needs to be kept at a minimum. A type I error will result in the rejection of a large number of good tires and is therefore costly. A type II error of passing a faulty batch of tires can result in fatal accidents on the roads, and therefore is extremely costly. (For purposes of this case, the probability of type II error, ?, is always calculated at ? = 2,790 psi.) It is believed that ? should be at most 1%. Currently, the company conducts the test with a sample size of 40 and an ? of 5%.
Barbara Bright is the purchasing agent for West Valve Company. West Valve sells industrial valves and fluid control devices. One of the most popular valves is the Western, which has an annual demand of 4,000 units. The cost of each valve is $90, and the inventory carrying cost is estimated to be 10% of the cost of each valve. Barbara has made a study of the costs involved in placing an order for any of the valves that West Valve stocks and she has concluded that the average ordering cost is $25 per order.,Furthermore, it takes about two weeks for an order to arrive from the supplier, and during this time the demand per week for West valves is approximately 80.,(a) What is the EOQ? ,(b) What is the ROP? ,(c) What is the average inventory? What is the annual holding cost? ,(d) How many orders per year would be placed? What is the annual ordering cost?
1. Conduct a hypothesis test for each sample at the .01 level of significance and determine what action, if any, should be taken. Provide the test statistic and p-value for each test.,2. Compute the standard deviation for each of the four samples. Does the assumption of .21 for the population standard deviation appear reasonable?,3. Compute limits for the sample mean x around ? = 12 such that, as long as a new sample mean is within those limits, the process will be considered to be operating satisfactorily. If x exceeds the upper limit or if x is below the lower limit, corrective action will be taken. These limits are referred to as upper and lower control limits for quality control purposes.,4. Discuss the implications of changing the level of significance to a larger value. What mistake or error could increase if the level of significance is increased?,Quality Associates, Inc., a consulting firm, advises its clients about sampling and statistical procedures that can be used to control their manufacturing processes. In one particular application, a client gave Quality Associates a sample of 800 observations taken during a time in which that client’s process was operating satisfactorily. The sample standard deviation for these data was .21; hence, with so much data, the population standard deviation was assumed to be .21. Quality Associates then suggested that random samples of size 30 be taken periodically to monitor the process on an ongoing basis. By analyzing the new samples, the client could quickly learn whether the process was operating satisfactorily. When the process was not operating satisfactorily, corrective action could be taken to eliminate the problem. The design specification indicated the mean for the process should be 12. The hypothesis test suggested by Quality Associates follows.,H0: ? = 12,H0: ? = 12,Corrective action will be taken any time h0 isrejected.