Question

practice makes perfect
63.1 pham is conducting a survey for the school newspaper. he surveyed 175 students at his school and found that 28 students had the new xphone, 44 students had a xpad computer, and 110 students had neither. pham then made a venn diagram to illustrate the information.
(image of venn diagram: two overlapping circles labeled xphone (21 in non-overlapping part) and xpad (37 in non-overlapping part), overlapping part with ?, and neither 110 below)

a. how did pham determine the number of students who have both xphones and xpads?

b. how many students have xphones or xpads? how did he find this?

c. recall that 28 students had the new xphone and 44 students had a xpad. why cannot pham just add 28 and 44 to get the answer to part (b)? explain

63.2 raul said he could not figure out how to put the data from problem 3-1 into a venn diagram because it was hard to find the number of students with both phones and pads, so he put the information into a table like the one below.

(table: rows labeled xphone, no xphone; columns labeled xpad, no xpad, totals. xphone row: xpad column 7, no xpad column ?, total 28. no xphone row: xpad column ?, no xpad column 110, total ?. totals row: xpad column 44, no xpad column ?, total 175)

a. complete raul’s table to figure out the number of students in the missing cells.

b. this type of table is called a two - way frequency table and is often used to organize information and calculate the probabilities. how is it similar to and different from a venn diagram? what are its advantages or

Explanation:

For Problem #3.1:

Step1: Find total with device

Total students with device: $175 - 110 = 65$

Step2: Calculate overlap (both devices)

Let $x$ = students with both. Sum of single devices + overlap = total with device: $21 + 37 + x = 65$
$x = 65 - 21 - 37 = 7$

Step3: Verify total iPhone count

Total iPhone users: $21 + 7 = 28$ (matches given)

Step4: Verify total iPad count

Total iPad users: $37 + 7 = 44$ (matches given)

Step5: Calculate "either/or" count

Total with iPhone or iPad: $28 + 44 - 7 = 65$ (or $175 - 110 = 65$)

Step1: Fill "no iPad, iPhone" cell

Students with iPhone but no iPad: $28 - 7 = 21$

Step2: Fill "iPad, no iPhone" cell

Students with iPad but no iPhone: $44 - 7 = 37$

Step3: Fill "no iPhone, total" cell

Total students with no iPhone: $175 - 28 = 147$

Step4: Fill "no iPad, total" cell

Total students with no iPad: $21 + 110 = 131$

Answer:

a. Pham first calculated total students with at least one device ($175-110=65$). Then he subtracted the students with only an iPhone (21) and only an iPad (37) from this total: $65 - 21 - 37 = 7$. So 7 students have both.
b. 65 students have an iPhone or an iPad. This is found by subtracting the number of students with neither (110) from the total surveyed (175), or by adding total iPhone, total iPad, then subtracting the overlap (to avoid double-counting): $28 + 44 - 7 = 65$.
c. Adding 28 and 44 counts the 7 students who have both devices twice. To get the correct number of unique students with either device, the overlap must be subtracted once to avoid double-counting.

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For Problem #3.2: