Chapter 5 Section 2 Please answer questions a through e Trevor is interested in

Question

Chapter 5 Section 2

Please answer questions a through e

Trevor is interested in purchasing the local hardware/sporting goods store in the small town of Dove Creek, Montana. After examining accounting records for the past several years, he found that the store has been grossing over $850 per day about 65% of the business days it is open. Estimate the probability that the store will gross over $850 for the following. (Round your answers to three decimal places.) (a) at least 3 out of 5 business days (b) at least 6 out of 10 business days (c) fewer than 5 out of 10 business days (d) fewer than 6 out of the next 20 business days If the outcome described in part (d) actually occurred, might it shake your confidence in the statement p = 0.65? Might it make you suspect that p is less than 0.65? Explain. Yes. This is unlikely to happen if the true value of p is 0.65. Yes. This is likely to happen if the true value of p is 0.65. No. This is unlikely to happen if the true value of p is 0.65. No. This is likely to happen if the true value of p is 0.65. (e) more than 17 out of the next 20 business days If the outcome described in part (e) actually occurred, might you suspect that p is greater than 0.65? Explain. Yes. This is unlikely to happen if the true value of p is 0.65. Yes. This is likely to happen if the true value of p is 0.65. No. This is unlikely to happen if the true value of p is 0.65. No. This is likely to happen if the true value of p is 0.65.

Explanation / Answer

a)

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 =    5      
p = the probability of a success =    0.65      
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   ) =    0.235169375
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.764830625 [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 =    10      
p = the probability of a success =    0.65      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.248504491
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.751495509 [ANSWER]

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c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    10      
p = the probability of a success =    0.65      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.09493408
          
Which is also          
          
P(fewer than   5   ) =    0.09493408 [ANSWER]

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d)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.65      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.000310575
          
Which is also          
          
P(fewer than   6   ) =    0.000310575 [ANSWER]

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Part d is a small probability, so

OPTION A. Yes, it is unlikely to happen is p = 0.65. [ANSWER]

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e)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.65      
x = our critical value of successes =    17      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   17   ) =    0.987882293
          
Thus, the probability of at least   18   successes is  
          
P(more than   17   ) =    0.012117707 [ANSWER]

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This is also a small probability, so

OPTION A: Yes, this is unlikely if p = 0.65. [ANSWER]

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