12-5 Samples and Surveys

Quick Review

You can obtain information about a population of people by surveying a smaller part of it, called a sample. The sample should be representative of the population. An unrepresentative sample or a poorly worded question can result in bias.

Example

A survey asks, “Should Plainville make itself proud by building a beautiful new library?” Is the question biased?

The question is biased. The words proud and beautiful make it clear that the answer is expected to be yes.

Exercises

Determine whether the sampling method is random, systematic, or stratified. Tell whether the method will give a good sample. Then write an unbiased survey question for the situation.

20. Movies An interviewer outside a movie theater asks every third person in line whether he or she will see more or fewer movies in the coming year.
21. Student Government Ten randomly chosen students in each class (freshman, sophomore, junior, and senior) are asked whom they support for student council president.

12-6 Permutations and Combinations

Quick Review

If there are m ways to make a first selection and n ways to make a second selection, then there are m · n m middle dot n  ways to make the two selections.

A permutation is an arrangement of objects in a specific order. The number of permutations of n objects arranged r at a time,   n P r , sub n , cap p sub r , comma  equals n ! ( n − r ) ! fraction n factorial , over open , n minus r , close . factorial end fraction

A combination is a selection of objects without regard to order. The number of combinations of n objects chosen r at a time,   n C r , sub n , cap c sub r , comma  equals n ! r ! ( n − r ) ! fraction n factorial , over r factorial . open , n minus r , close . factorial end fraction

Example

In how many ways can you choose 3 people to serve on a committee out of a group of 7 volunteers?

The order does not matter, so this is a combination problem.

  7 C 3 = 7 ! 3 ! ( 7 − 3 ) ! = 7 ! 3 ! 4 ! Write using factorials .   = 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 ( 3 ⋅ 2 ⋅ 1 ) ( 4 ⋅ 3 ⋅ 2 ⋅ 1 ) Write the factorials as products .   = 35 Simplify . table with 3 rows and 3 columns , row1 column 1 , sub 7 , cap c sub 3 , column 2 equals . fraction 7 factorial , over 3 factorial . open , 7 minus 3 , close . factorial end fraction . equals . fraction 7 factorial , over 3 factorial 4 factorial end fraction , column 3 cap writeusingfactorials . . , row2 column 1 , , column 2 equals . fraction 7 dot 6 dot 5 dot 4 dot 3 dot 2 dot 1 , over open . 3 dot 2 dot 1 . close . open . 4 dot 3 dot 2 dot 1 . close end fraction , column 3 table with 2 rows and 1 column , row1 column 1 , cap writethefactorialsas , row2 column 1 , products , . , end table , row3 column 1 , , column 2 equals 35 , column 3 cap simplify , . , end table

There are 35 ways to choose 3 people out of a group of 7.

Exercises

Find the number of permutations.

22.   9 P 5 sub 9 , cap p sub 5
23.   3 P 2 sub 3 , cap p sub 2
24.   8 P 3 sub 8 , cap p sub 3
25.   5 P 2 sub 5 , cap p sub 2
26.   6 P 4 sub 6 , cap p sub 4
27.   7 P 2 sub 7 , cap p sub 2

Find the number of combinations.

28.   8 C 2 sub 8 , cap c sub 2
29.   9 C 4 sub 9 , cap c sub 4
30.   5 C 3 sub 5 , cap c sub 3
31.   6 C 3 sub 6 , cap c sub 3
32.   7 C 3 sub 7 , cap c sub 3
33.   5 C 4 sub 5 , cap c sub 4
34. Side Dishes You can choose any 2 of the following side dishes with your dinner: mashed potatoes, cole slaw, french fries, applesauce, or rice. How many different combinations of side dishes can you choose?
35. Talent Show There are 8 groups participating in a talent show. In how many different orders can the groups perform?
36. Clothing You have 6 shirts, 7 pairs of pants, and 3 pairs of shoes. How many different outfits can you wear?

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