Up to e i have the answers. 1. According to the FDA \"Food Defect Action Levels\

Question

Up to e i have the answers. 1. According to the FDA "Food Defect Action Levels" booklet there is an average of 3 insect fragments per 10 grams of peanut butter. Suppose we let X equal the number of insect fragments in a randomly chosen 10 gram serving of peanut butter. It is reasonable that X is a Poisson random variable with a parameter of 3. a. What is the expected value of X? b. What is the variance of X? c. What is the standard deviation of X? WebAssign will check your answer for the correct number of significant figures. d. What is the variance of 4X? e. What is the probability that there will be no insect fragments in the serving? WebAssign will check your answer for the correct number of significant figures. f. What is the probability that there will be exactly 3 insect fragments in the serving? WebAssign will check your answer for the correct number of significant figures. g. What is the probability that no more than 2 insect fragments are in the serving? WebAssign will check your answer for the correct number of significant figures. h. What is the probability that more than 2 insect fragments are in the serving? WebAssign will check your answer for the correct number of significant figures. i. What is the probability that between 3 and 5 (3<=X<=5) insect fragments are in the serving? WebAssign will check your answer for the correct number of significant figures. j. We independently select 3 10-gram peanut butter servings. What is the probability that the maximum number of insect fragments in the 3 servings is <=2?WebAssign will check your answer for the correct number of significant figures.

Explanation / Answer

f. What is the probability that there will be exactly 3 insect fragments in the serving? WebAssign will check your answer for the correct number of significant figures.

Note that the probability of x successes is        
        
P(x) = u^x e^(-u) / x!        
        
where        
        
u = the mean number of successes =    3    
        
x = the number of successes =    3    
        
Thus, the probability is        
        
P (    3   ) =    0.224041808 [ANSWER]

**************************

g. What is the probability that no more than 2 insect fragments are in the serving? WebAssign will check your answer for the correct number of significant figures.

Using a cumulative poisson distribution table or technology, matching        
        
u = the mean number of successes =    3    
        
x = the maximum number of successes =    2    
        
Then the cumulative probability is        
        
P(at most   2   ) =    0.423190081 [ANSWER]

**************************

h. What is the probability that more than 2 insect fragments are in the serving? WebAssign will check your answer for the correct number of significant figures.

Note that P(more than x) = 1 - P(at most x).        
        
Using a cumulative poisson distribution table or technology, matching        
        
u = the mean number of successes =    3    
        
x = our critical value of successes =    2    
        
Then the cumulative probability of P(at most x) from a table/technology is        
        
P(at most   2   ) =    0.423190081
        
Thus, the probability of at least   3   successes is
        
P(more than   2   ) =    0.576809919 [ANSWER]

******************************

i. What is the probability that between 3 and 5 (3<=X<=5) insect fragments are in the serving? WebAssign will check your answer for the correct number of significant figures.

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)        
        
Here,        
        
x1 =    3    
x2 =    5    
        
Using a cumulative poisson distribution table or technology, matching        
        
u = the mean number of successes =    3    
        
        
Then        
        
P(at most    2   ) =    0.423190081
P(at most    5   ) =    0.916082058
        
Thus,        
        
P(between x1 and x2) =    0.492891977   [ANSWER]

**************************************************************  

j. We independently select 3 10-gram peanut butter servings. What is the probability that the maximum number of insect fragments in the 3 servings is <=2?WebAssign will check your answer for the correct number of significant figures.

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    3      
          
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.423190081
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.576809919

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 =    3      
p = the probability of a success =    0.576809919      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.075789046    [ANSWER]

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.