If the design load is considered to be the mean + 2 standard deviation value, what is the probability that it will be exceeded?

Statistics and Probability
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Prob 1.
The magnitude of a load acting on a structure can be modeled by a normal distribution with a mean of 100 kip and a standard deviation of 20 kip.
(a) If the design load is considered to be the 90th percentile value, determine the design load.
(b) If the design load is considered to be the mean + 2 standard deviation value, what is the probability that it will be exceeded?
(c) A load of magnitude less than zero is physically illogical: calculate its probability. Is a normal distribution appropriate to model the load?

Prob 2.
The annual rainfall for a city is assumed to be normally distributed with a mean of 100 cm, and its mean ±3 standard deviation values are estimated to be 160 and 40 cm. respectively.
(a) Calculate the standard deviation of the annual rainfall.
(b) What is the probability that the rainfall will be less than 0?
(c) What is the probability that the annual rainfall will be within the ±3 standard deviation values?
(d) Is normal distribution appropriate in this case?

Prob 3.
Solve problem 1 again using the lognormal assumption of the load.

Prob 4.
The compressive strength of concrete delivered by a supplier can be modeled by a lognor-mal random variable. Its mean and the coefficient of variation are estimated to be 4.7 ksi and 0.21, respectively.
(a) If the 10th percentile value is the design value, calculate the value of the compressive strength to be used in a design.
(b) Suppose the COV of the compressive strength is reduced to 0.10 without affecting its mean value by introducing quality control procedures. Calculate the design value of the compressive strength if it is assumed to be the 10th percentile value.
(c) By comparing the results obtained in Parts (a) and (b), discuss whether quality control measures are preferable.

Prob 5.
If the load S applied on a bar is assumed to be a normal distribution with a mean of 200 MPa and a standard deviation of 20 MPa. The bar’s ultimate strength R is also a normal distribution with a mean of 210 MPa with a standard deviation of 5 MPa. It is known that a summation or subtraction of two normal distribution also follows the normal distribution.
Please calculate the failure probability of the bar.
Hint: The failure happens when difference of the bar strength and the applied load is less than zero.

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