11-7 Samples and Surveys

Quick Review

A sample is part of a population. For a random sample, all members of the population are equally likely to be chosen. A bias is a systematic error introduced by the sampling method.

Example

Identify any bias in the survey question “Do you think the school day should be extended even longer than it already is?”. Explain.

There is bias because the question is leading. It implies that the school day is already too long and should not be extended.

Exercises

Determine if each of the following is a random sample. Explain your answer.

43. The first 50 names in the telephone directory
44. Twelve jurors chosen through examination by opposing lawyers
45. Two class representatives chosen by drawing names from a hat
46. Five newspapers picked on the basis of circulation size
47. The city council is trying to determine if the city's residents support the building of a new parking garage. They poll 200 people at the local bus station. Identify any bias in the sampling method.

11-8 Binomial Distributions

Quick Review

A binomial experiment has repeated independent trials with each trial having two possible outcomes. In a binomial experiment with probability of success p and of failure q (so p + q = 1), the probability of exactly x successes in n trials is   n C x p x q n − x . sub n , cap c sub x , p to the x . q super n minus x end super . .  This value is the binomial probability. The Binomial Theorem says that for every positive integer n, ( a + b ) n =   n C 0 a n +   n C 1 a n − 1 b +   n C 2 a n − 2 b 2 + … +   n C n − 1 a b n − 1 +   n C n b n . open eh plus b close to the n . equals , sub n , cap c sub 0 , eh to the n , plus , sub n , cap c sub 1 . eh super n minus 1 end super . b plus , sub n , cap c sub 2 . eh super n minus 2 end super . b squared , plus dot dot dot plus , sub n . cap c sub n minus 1 end sub . eh b super n minus 1 end super . plus , sub n , cap c sub n , b to the n , .

Example

In a binomial trial, the probability of success is 0.8 for each trial. Find the probability of exactly 4 successes in 7 trials.

p = 0.8 , p equals 0.8 comma   q = 0.2 , q equals 0.2 comma   x = 4 , x equals 4 comma  and n = 7 n equals 7

P ( 4 ) =   7 C 4 ( 0.8 ) 4 ( 0.2 ) 3   = 7 ! 3 ! 4 ! ( 0.8 ) 4 ( 0.2 ) 3   ≈ 0.115 table with 3 rows and 2 columns , row1 column 1 , p open 4 close , column 2 equals , sub 7 , cap c sub 4 . open 0.8 close to the fourth . open 0.2 close cubed , row2 column 1 , , column 2 equals . fraction 7 factorial , over 3 factorial 4 factorial end fraction . open 0.8 close to the fourth . open 0.2 close cubed , row3 column 1 , , column 2 almost equal to , 0.115 , end table

The probability of 4 successes in 7 trials is 0.115.

Exercises

For each of the following binomial experiments, state the value of p, the probability of success.

48. A series of coin flips, where success is “heads.”
49. A series of number cube rolls, where success is “2 or 4.”

In a binomial trial, the probability of success is 0.6 for each trial. Find the probability of each of the following.

50. 13 successes in 24 trials
51. 9 successes in 20 trials
52. 9 successes in 15 trials
53. 6 failures in 12 trials

Use the Binomial Theorem to write each of the following.

54. the third term in the expansion of ( a + b ) 7 open eh plus b close to the seventh
55. the sixth term in the expansion of ( a + b ) 8 open eh plus b close to the eighth

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