b. (5 points) Use the frequency distribution on the previous page to construct a histogram for these data.1. An article in Technometrics ["Validation of Regression Models: Methods and Examples" (1977, Vol. 19(4), p. 425)] presented the following data (which have been rounded to the nearest whole value) on the motor fuel octane ratings of several blends of gasoline: 89 99 90 92 93 88 88 91 95 88 90 83 88 93 88 90 84 90 92 91 93 94 88 92 90 91 91 88 94 97 89 90 89 91 89 90 84 92 92 90 92 88 93 91 93 89 92 94 87 87 89 90 90 91 85 91 94 89 93 90 87 90 91 91 93 93 92 93 80 100 90 89 87 90 96 91 88 92 a. (5 points) Construct a frequency distribution for these data by filling in the table below. Use 8 bins. Lower limit Upper limit Midpoint Frequency Relative frequencyAn article in Transactions of the Institution of Chemicni Engineers {1956, Vol. 34, pp. 230293} reported data from an experiment investigating the effect of several process variables on the vapor phase oxidation of naphthalene. A sample of the percentage mole conversion of naphthalene to maleic anhydride follows: 4.2, 4.7, 4.7, 5.0, 3.8, 3.6, 3.0, 5.1, 3.1, 3.8, 4.8, 4.0, 5.2, 4.3, 2.8, 2.0, 2.3, 3.3, 4.3, 5.0. a. [5 points] Calculate the sample mean, sample variance, and sample standard deviation. Feel free to use your calculator. b. (5 points) Construct a box plot of the data, and mark the location of outliers (if any} with an asterisk. 3. {5 points] Using your calculator, construct a normal probability plot for the octane data from problem 1, and make a sketch of that plot below. From the plot you sketched, does it seem reasonable to assume that octane rating is normally distributed? 4. {5 points] A civil engineer is analyzing the compressive strength of concrete. Compressive strength is normally distributed with (11'2 =1000 psi: .A random sample of 12 specimens has a mean compressive strength of E =3250 psi. Construct a 95% twosided confidence interval on mean compressive strength. Comment on whether a 99% twosided confidence interval would be wider or narrower than the one you found. 5. An article in the Journal ofC'omposite Materials [December 1989, Vol. 23(12}, pp. 12001215] describes the effect of delamination on the natural frequency of beams made from composite laminates. Five such delaminated beams were subjected to loads, and the resulting frequencies {in hertz} were as follows: 230.55, 233.05, 232.53, 223.43, 232.53, 230.55, 233.05, 232.53, 223.43, 232.53 a. {5 points] Check the assumption of normality in the population. Show work to support your conclusion. b. (5 points) Calculate a 90% two-sided confidence interval on mean natural frequency.6. An article in The Engineer ("Redesign for Suspect Wiring," June 1990] reported the results of an investigation into wiring errors on commercial transport aircraft that may display faulty information to the ight crew. Such a wiring error may have been responsible for the crash of a British Midland Airways aircraft in January 1989 by causing the pilot to shut down the wrong engine. Of 1600 randomly selected aircraft, 8 were found to have wiring errors that could display incorrect information to the flight crew. a. (5 points) Find a 95% confidence interval on the proportion of aircraft that have such wiring errors. b. [5 points] Suppose that you use the information in this example to provide a preliminary estimate of ,0. How large a sample would be required to produce an estimate ofp that we are 95% condent differs from the true 1uralue by at most 0.003