Question 1150657
<i>a. What is the probability that more than 20% of the business travelers say that the reason for their most recent business trip was an internal company visit?</i>
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Since there are 950 travelers selected, we need to figure out how many travelers 20% equals.  So...(0.20) * 950 = 190.  Therefore, the question is really asking for the probability that MORE than 190 travelers say the reason for their most recent business trip was an internal company visit.
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n = 950
p = 0.19
q = 0.81
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n*p = 950*(0.19) = 180.5
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SD = {{{sqrt(n*p*q)}}} = {{{sqrt(950*(0.19)*(0.81))}}} = 12.09
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Since the question asks for MORE than 190 travelers (it actually says MORE than 20%)...meaning, NOT including a result of 190 travelers...we find the probability that more than 190.5 travelers say the reason for their most recent business trip was for an internal company visit.
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We do this by finding the probability of FEWER than 190.5 travelers, then subtracting this result from 1.
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Z-score = {{{(190.5 - 180.5)/12.09}}} = {{{10/12.09}}} = 0.83
<pr>
Look up 0.83 on a z-table.  The result is 0.7967.  (This is the probability that FEWER than 190.5 travelers say the reason for their most recent business trip was for an internal company visit.)
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Subtracting this result from 1, we get {{{1 - 0.7967}}} = 0.2033
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So, there is a 0.2033 probability that more than 190 randomly selected business travelers (or, more than 20%) say the reason for their most recent business trip was an internal company visit.
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<i>b. What is the probability that between 18% and 20% of the business travelers say that the reason for their most recent business trip was an internal company visit?</i>
<pr>
We need to figure out what number of business travelers 18% and 20% of 950 are:
<pr>
(0.18) * 950 = 171
(0.20) * 950 = 190
<pr>
So, we want to find out the probability that between 171 and 190 business travelers say that the reason for their most recent business trip was an internal company visit. 
<pr>
First, figure out the probability that 190 business travelers or fewer say that the reason for their most recent business trip was an internal company visit.  Since this includes a result of 190 travelers, we do this by calculating the probability that FEWER than 190.5 business travelers or fewer say that the reason for their most recent business trip was an internal company visit.
<pr>
Fortunately, we already did this in the first part of the problem.  We found that there is a 0.7967 probability that fewer than 190.5 business travelers say that the reason for their most recent business trip was an internal company visit.
<pr>
Now, we need to figure out the probability that 171 business travelers or fewer say that the reason for their most recent business trip was an internal company visit.  This includes a result of 171 travelers, so we do this by calculating the probability that fewer than 171.5 business travelers say that the reason for their most recent business trip was an internal company visit.
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Just to refresh on the numbers we calculated earlier:
<pr>
n = 950
p = 0.19
q = 0.81
<pr>
n*p = 950*(0.19) = 180.5
<pr>
SD = {{{sqrt(n*p*q)}}} = {{{sqrt(950*(0.19)*(0.81))}}} = 12.09
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Z-score = {{{(171.5 - 180.5)/12.09}}} = {{{(-9)/12.09}}} = -0.74
<pr>
Look up -0.74 on a z-table.  The result is 0.2296.  So, there is 0.2296 probability that fewer than 171 randomly selected business travelers (or, fewer than 18%) say the reason for their most recent business trip was an internal company visit.
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To find the probability that between 171 and 190 business travelers say the reason for their most recent business trip was an internal company visit, we simply subtract the two probabilities from one another:
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{{{0.7967 - 0.2296}}} = 0.5671
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So, there is a 0.5671 probability that between 171 and 190 randomly selected business travelers (or, between 18% and 20%) say the reason for their most recent business trip was an internal company visit.