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- For a certain type of salamander, the mass X of a randomly chosen adult is normally distributed. 95% of the adults in this species have a mass between 6.7 grams and 8.1 grams.Suppose that samples of polythene bags from two manufacturers A and B are tested by a prospective buyer for bursting pressure, with the following results: If the prices are the same, which manufacture’s bags would be preferred by the buyer? Why?The specification for the pull strength of a wire that connects an integrated circuit to its frame is 10 g or more. Units made with aluminum wire have a defect rate of 10%. A redesigned manufacturing process, involving the use of gold wire, is being investigated. The goal is to reduce the rate of defects to 5% or less. Out of the first 100 units manufactured with gold wire, only 4 are defective. True or false: a) Since only 4% of the 100 units were defective, we can conclude that the goal has been reached. b) Although the sample percentage is under 5%, this may represent sampling variation, so the goal may not yet be reached. c) There is no use in testing the new process, because no matter what the result is, it could just be due to sampling variation. d) If we sample a large enough number of units, and if the percentage of defective units is far enough below 5%, then it is reasonable to conclude that the goal has been reached.
- Refer to Exercise 6. Assume that c = 448 J/kg°C and ΔQ = 1210 J are known with negligible uncertainty. Assume the mass is m = 0.75 ± 0.01 kg. Estimate ΔT, and find the relative uncertainty in the estimate.A metal rod of length L and a sphere of radius R are to be joined to form a small automobile part. L and R are normally distributed with the following parameters: E(L) = 2cm, E(R) = 0.5cm, V(:) =0.1^2 cm, V(R)=0.05^2 cm^2. Assume that L and R are independent. What is the probability that the total length of a single assembly is less than 2.90 cm?Consider the following 10 measurements about the cohesion of a soil: 12 kPa, 14 kPa, 15 kPa, 14.5 kPa, 16.0 kPa, 18 kPa, 15 kPa, 16.2 kPa, 17. 6 kPa, 13 kPa. Work out the solutions of the following two problems, assuming the cohesion follows the normal and lognormal distributions, respectively. (1) Draw the PDF of the cohesion; (2) Evaluate the probability that the cohesion based on the PDF is less than 5 kPa, 10 kPa, and 15 kPa; (3) Assess the effect of the type of distribution on the results.
- In 58 specimens of a new aluminum alloy that is developed as a material for the next generation of aircraft, the compressive strength was measured, obtaining the following results: 66.4 67.7 68.0 68.0 69.5 70.4 71.6 72.9 69.0 69.169.2 69.3 69.3 69.5 70.3 71.5 72.7 68.9 69.9 70.070.0 70.1 70.2 70.3 71.3 72.6 68.8 69.8 70.8 70.971.0 71.1 71.2 71.3 72.4 68.6 69.8 70.6 71.8 71.871.9 72.1 72.2 72.3 68.4 69.7 70.6 71.7 73.3 73.574.2 74.5 75.3 68.3 69.6 70.5 71.6 73.1 a. Find a 95% confidence interval for the mean strength of the aluminum alloy.b. If you want to test Ho: Mu = 72 against H1: Mu different from 72 with a significance level of 0.05, what is the probability of a type II error for Mu = 71?In 58 specimens of a new aluminum alloy that is developed as a material for the next generation of aircraft, the compressive strength was measured, obtaining the following results: 66.4 67.7 68.0 68.0 69.5 70.4 71.6 72.9 69.0 69.169.2 69.3 69.3 69.5 70.3 71.5 72.7 68.9 69.9 70.070.0 70.1 70.2 70.3 71.3 72.6 68.8 69.8 70.8 70.971.0 71.1 71.2 71.3 72.4 68.6 69.8 70.6 71.8 71.871.9 72.1 72.2 72.3 68.4 69.7 70.6 71.7 73.3 73.574.2 74.5 75.3 68.3 69.6 70.5 71.6 73.1 a. Find a 95% confidence interval for the mean resistance of thealuminium alloy.b. If you want to test Ho:Mu =72 against H1: Mu different from 72 with a significance level of 0.05, what is the probability of a type II error for Mu= 71?If X has a uniform density with α = 0 and β = 1, show that the random variable Y = −2. In X has a gamma dis-tribution. What are its parameters?
- Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?A point D is chosen on the line AB, whose midpointis C and whose length is a. If X, the distance from D toA, is a random variable having the uniform density withα = 0 and β = a, what is the probability that AD, BD,and AC will form a triangle?To investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in a table below. The researchers concluded that swimming in guar syrup does not change swimming speed. (Use a statistical computer package to calculate the P-value. Use ?d = ?water − ?guar syrup. Round your test statistic to two decimal places and your P-value to three decimal places.) Swimmer Velocity (m/s) Water Guar Syrup 1 1.43 1.23 2 1.02 1.96 3 1.37 1.54 4 1.23 1.76 5 1.42 1.07 6 1.09 1.63 7 1.56 1.31 8 1.33 1.58 9 1.96 1.30 10 1.89 1.23 11 1.15 1.15 12 1.11 2.00 13 1.31 1.59 14 1.79 1.23 15 1.12 1.05 16 1.84 1.12 17 1.72 1.42 18 1.99 1.79 19 1.53 1.80 20 0.92 1.69 t = df = P-value = Is there sufficient evidence to suggest that…
