This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's the solution to question 6. b): Let U be the set of all people surveyed (representing 100%). Let C be the set of people using cellular data. Let W be the set of people using Wi-Fi. Given information in percentages: People using cellular data but not Wi-Fi: n(C W) = 20\% People using Wi-Fi but not cellular data: n(W C) = 65\% People using neither cellular data nor Wi-Fi: n(C W)' = 5\% Step 1: Calculate the percentage of people using both cellular data and Wi-Fi. The sum of all disjoint regions must equal 100%: n(U) = n(C W) + n(W C) + n(C W) + n(C W)' 100\% = 20\% + 65\% + n(C W) + 5\% 100\% = 90\% + n(C W) n(C W) = 100\% - 90\% n(C W) = 10\% b) (i) If C and W denote the sets of people who are using cellular data and Wi-Fi respectively, write the cardinality of n(C W). The cardinality of n(C W) is the percentage of people using at least one of the services. n(C W) = n(C W) + n(W C) + n(C W) n(C W) = 20\% + 65\% + 10\% n(C W) = 95\% The cardinality of n(C W) is 95\%. b) (ii) Represent the above information in a Venn-diagram. A Venn diagram would illustrate: • A rectangle representing the universal set U (100%). • Two overlapping circles, one labeled C and one labeled W. • The region for C only (C W) contains 20%. • The region for W only (W C) contains 65%. • The intersection region (C W) contains 10%. • The region outside both circles ((C W)') contains 5%. b) (iii) Find the percent of people who are using Wi-Fi. The percent of people using Wi-Fi is the sum of those using Wi-Fi only and those using both: n(W) = n(W C) + n(C W) n(W) = 65\% + 10\% n(W) = 75\% The percent of people using Wi-Fi is 75\%. b) (iv) By what percent of the people who are not using cellular data is more or less than the people who are not using Wi-Fi? First, find the percent of people who are not using cellular data (n(C')): n(C') = n(W C) + n(C W)' n(C') = 65\% + 5\% n(C') = 70\% Next, find the percent of people who are not using Wi-Fi (n(W')): n(W') = n(C W) + n(C W)' n(W') = 20\% + 5\% n(W') = 25\% Now, find the difference: n(C') - n(W') = 70\% - 25\% n(C') - n(W') = 45\% Since the result is positive, the percentage of people not using cellular data is 45% more than the percentage of people not using Wi-Fi. It is 45% more. What's next?