ChE 311.3 – Mathematical Modelling I Assignment # 3 Posted: Due: October 16th October 23rd (beginning of class) The following questions are taken from “Statistics and Probability for Engineering Applications” by W. J. DeCoursey. 1. A manufacturer produces a special alloy steel with an average tensile strength of 25,800 psi. The standard deviation of the tensile strength is 300 psi. Strengths are approximately normally distributed. A change in composition of the alloy is tried in an attempt to increase its strength. A sample consisting of eight specimens of the new composition is tested. Unless an increase in the strength is significant at the 1% level, the manufacturer will return to the old composition. Standard deviation is not affected. a) b) c) If the mean strength of the sample of eight items is 26,100 psi, should the manufacturer continue with the new composition? What is the minimum mean strength that will justify continuing with the new composition? How large would the true mean strength of the new composition (i.e., a new population mean) have to be to make the odds 9 to 1 in favor of obtaining a sample mean at least as big as the one specified in part (b)? 2. Insulators produced by a factory have a breakdown voltage distribution that can be approximated by a normal distribution. The coefficient of variation is 5%. a) b) What is the smallest sample size that will ensure probability of 90% that the sample mean measured is between 0.98 times the population mean and 1.02 times the population mean? If the sample size is 40, what is now the confidence level associated with the sample mean lying between 0.98 times the population mean and 1.02 times the population mean? 3. Jack Spratt is in charge of quality control of the concrete poured during the construction of a certain building. He has specimens of concrete tested to determine whether the concrete strength is within the specifications; these call for a mean concrete strength of no less than 30 MPa. It is known that the strength of such specimens of concrete will have a standard deviation of 3.8 MPa and that the normal distribution will apply. Mr. Spratt is authorized to order the removal of concrete which does not meet specifications. Since the general contractor is a burly sort, Mr. Spratt would like to avoid removing the concrete when the action is not justified. Therefore, the probability of rejecting the concrete when it actually meets the specification should be no more than 1%. What size sample should Mr. Spratt use if a sample mean 10% less than the specified mean strength will cause rejection of the concrete pour? State the null hypothesis and alternative hypothesis. (Instructor’s hint: 30 MPa.) 4. Carbon composition resistors with mean resistance 560 Ω and coefficient of variation 10% are produced by a factory. They are sampled each hour in the quality control lab. What sample size would be required so that there is 95% probability that the mean resistance of the sample lies within 10 Ω of 560 Ω if the population mean has not changed?