Underage drinking, Part I: Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in 2008. (a) Suppose a random sample of the ten 18-20 year olds is taken. Is the use of the binomial distribution appropriate for calculating the probability that exactly six consumed alcoholic beverages? No, this follows the bimodal distribution No, the normal distribution should be used Yes, there are 10 independent trials, each with exactly two possible outcomes, and a constant probability associated with each possible outcome No, the trials are not independent (please round to four (b) Calculate the probability that exactly 6 out of 10 randomly sampled 18-20 year olds consumed an alcoholic drink. (please round to four decimal places) (c) What is the probability that exactly four out of the ten 18-20 year olds have not consumed an alcoholic beverage? decimal places) (d) What is the probability that at most 2 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages? (please round to four decimal places) (e) What is the probability that at least 1 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages? (please round to four decimal places)

a. Yes, there are 10 independent trials, each with exactly two possible outcomes, and a constant probability associated with each possible outcome.

b. The probability is 0.2029 or 20.29%

c. The probability is 0.2029 or 20.29%

d. The probability is 0.1671 or 16.71%

e. The probability is 0.9975 or 99.75%

Step-by-step explanation:

a. Yes, there are 10 independent trials, each with exactly two possible outcomes, and a constant probability associated with each possible outcome.

b. Let's use the binomial distribution table, this way:

Binomial distribution (n=10, p=0.697)

 f(x)F(x)1 - F(x)

xPr[X = x]Pr[X ≤ x]

00.00000.0000

10.00020.0002

20.00160.0017

30.00950.0112

40.03840.0496

50.10590.1555

60.2029 0.3584

70.26680.6252

80.23010.8553

90.11760.9729

100.02711.0000

The probability is 0.2029 or 20.29%

c. If 69.7% of 18-20 years old consumed alcoholic beverages in 2008, therefore, 30.3% did not and the binomial distribution table is:

Binomial distribution (n=10, p=0.303)

 f(x)F(x)1 - F(x)

xPr[X = x]Pr[X ≤ x]

00.02710.0271

10.11760.1447

20.23010.3748

30.26680.6416

40.2029 0.8445

50.10590.9504

60.03840.9888

70.00950.9983

80.00160.9998

90.00021.0000

100.00001.0000

The probability is 0.2029 or 20.29%

d. Let's use the binomial distribution table, this way:

Binomial distribution (n=5, p=0.697)

 f(x)F(x)1 - F(x)

xPr[X = x]Pr[X ≤ x]

00.00260.0026

10.02940.0319

20.13510.1671

30.31090.4779

40.35760.8355

50.16451.0000

P(0) + P(1) + P (2) = 0.0026 + 0.0294 + 0.1351

The probability is 0.1671 or 16.71%

e. Using the same binomial distribution table we used in d. we have:

P(1) + P (2) + P(3) + P(4) + P (5) = 0.0294 + 0.1351 + 0.3109 + 0.3576 + 0.1645

The probability is 0.9975 or 99.75%