(04.02 MC)  To increase sales, an online clothing store began giving a 50% off coupon to random customers. Customers didn’t know whether they would receive the coupon until after the final sale. The website claimed that one in five customers received the coupon. Six customers each made purchases from the website. Let X = the number of customers that received the 50% off coupon.Part A: Is X a binomial random variable? Explain. (3 points)Part B: What is the mean and standard deviation of X? Provide an interpretation for each value in context. (4 points)Part C: Two of the six customers receive the coupon with their purchase. Is the store’s claim accurate? Compute P(X ≥ 2) and use the result to justify your answer. (3 points) (10 points)

The graph of f is given. Sketch the graph of the function y = –f(x) + 3. ​   ​

(04.06 HC)  The air force reports that the distribution of heights of male pilots is approximately normal, with a mean of 72.6 inches and a standard deviation of 2.7 inches. Part A: A male pilot whose height is 74.2 inches is at what percentile? Mathematically explain your reasoning and justify your work. (5 points) Part B: Air force fighter jets can accommodate heights of soldiers between 70 inches and 78 inches without compromising safety. Anyone with a height outside that interval cannot fly the fighter jets. Describe what this interval looks like if displayed visually. What percent of male pilots are unable to fly according to this standard? Show your work and mathematically justify your reasoning. (5 points) (10 points)

A beam has dimensions of b = 14 in., h = 30 in., d’ = 2.5 in., and d = 27.5 in. It is reinforced with 2 No. 5 bars on the compression side and 4 No. 8 bars on the tension side. The concrete strength is 3,700 psi, and the yield strength of the reinforcement is 60,000 psi. If the depth of the neutral axis is c = 4.481 in., determine the force in the compression reinforcement, Cs.

(06.07 MC)  The number of cavities in school-aged children and size of vocabulary are known to have a strong negative association. Toby concluded that the number of cavities causes a decrease in vocabulary. Is Toby’s conclusion valid? Explain. (4 points)

The basic strength requirement from ACI 318 is “Reduced nominal strength ​≤ Factored load effects.”

A beam has dimensions of b = 18 in., h = 22 in., and d = 19.5 in. and is reinforced with 3 No. 8 bars. The concrete strength is 4,600 psi, and the yield strength of the reinforcement is 60,000 psi. If the depth of the compressive stress block is a = 2.02046 in. and the strain in the reinforcement is εs = 0.020742, determine the strength φMn for this beam. Assume the transverse reinforcement is not spirals.

A beam has dimensions of b = 14 in., h = 30 in., and d = 27.5 in. and is reinforced with 4 No. 6 bars. The concrete strength is 9,400 psi, and the yield strength of the reinforcement is 60,000 psi. If the depth to the neutral surface is c = 1.452365 in., determine the strain in the reinforcement.

A beam has dimensions of b = 16 in., h = 30 in., and d = 27.5 in. and is reinforced with 4 No. 6 bars. The concrete strength is 5,800 psi, and the yield strength of the reinforcement is 60,000 psi. Determine the nominal moment capacity, Mn, of the beam.

(04.05 LC)  Daphne’s teacher told her that the standardized score (z-score) for her history exam, as compared to the exam scores of other students in the course, is 2.50. Which of the following is the best interpretation of this standardized score? (4 points)