Please help thanks Problem 4 A supplier claims that at most

Please help :( thanks

Problem 4. A supplier claims that at most 5% of the concrete beams fail the strength test. To test the claim you, as a quality-control engineer, randomly select 25 beams and run the strength test. (a) What is the probability that 2 beams in the sample fail the test? (b) What is the probability that at least 4 beams fail the test? (c) What is the probability that between 2 and 5 beams fail the test? (d) What is the probability that at most 2 beams fail the test?

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    25      
p = the probability of a success =    0.05      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.230517651 [ANSWER]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.05      
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.965909399
          
Thus, the probability of at least   4   successes is  
          
P(at least   4   ) =    0.034090601 [ANSWER]

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c)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    2      
x2 =    5      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.05      
          
Then          
          
P(at most    1   ) =    0.642375854
P(at most    5   ) =    0.998787039
          
Thus,          
          
P(between x1 and x2) =    0.356411185   [ANSWER]

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d)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.05      
x = the maximum number of successes =    2      
          
Then the cumulative probability is          
          
P(at most   2   ) =    0.872893504 [ANSWER]

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