91Ó°ÊÓ

4.17 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 any given year. (a) Suppose a random sample of 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? Explain. (b) Calculate the probability that exactly 6 out of 10 randomly sampled 18- 20 year olds consumed an alcoholic drink. (c) What is the probability that exactly four out of ten \(18-20\) year olds have not consumed an alcoholic beverage? (d) What is the probability that at most 2 out of 5 randomly sampled \(18-20\) year olds have consumed alcoholic beverages? (e) What is the probability that at least 1 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?

Step by step solution

01

Verify Binomial Distribution Conditions

For the binomial distribution to be appropriate, certain conditions must be met: fixed number of trials, each trial is independent, there are only two possible outcomes (success and failure), and the probability of success is the same for each trial. In this problem, we have 10 individuals, each with two outcomes (consumed or did not consume), and the probability of consumption is 69.7%. Therefore, using the binomial distribution is appropriate.

02

Calculate Probability for Part (b)

To calculate the probability of exactly 6 out of 10 individuals consuming alcoholic beverages, use the binomial probability formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(n = 10\), \(k = 6\), \(p = 0.697\). Substitute these values:\[ P(X=6) = \binom{10}{6} (0.697)^6 (1-0.697)^{4} \].Calculate this probability.

03

Calculation for Part (c)

To find the probability that exactly four have not consumed alcohol, we need to calculate the probability that exactly six have consumed:\[ P(X=6) \] from Step 2. Since this is the same scenario talked about inversely, the result will apply here too. Hence there is no new calculation.

04

Calculation for Part (d)

To find the probability that at most 2 out of 5 have consumed alcoholic beverages, calculate\[ P(X \leq 2) = P(X=0) + P(X=1) + P(X=2) \]where n=5 and p=0.697. Use the binomial formula for each probability and add them together.

05

Calculation for Part (e)

The probability that at least 1 out of 5 have consumed can be calculated by finding the complement of none having consumed (\(X=0\)):\[ P(X \geq 1) = 1 - P(X=0) \].Use the binomial formula for \(P(X=0)\) and subtract from 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation

When working with the binomial distribution, probability calculation is key to understanding expected outcomes. The binomial distribution is used in statistics to determine the probability of a certain number of successes in a fixed number of independent trials, given a specific probability of success in each trial.

The binomial probability formula is:

• \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \)

In this formula:

• \( n \) is the total number of trials.
• \( k \) is the number of successful trials you're calculating the probability for.
• \( p \) is the probability of success in each trial.
• The formula \( \binom{n}{k} \) represents the number of combinations of \( n \) trials taken \( k \) at a time.

This formula helps calculate the likelihood of a given number of successes in a sample. It's essential to plug in the correct values for your specific problem to get accurate results.

Sampling

Sampling refers to the process of selecting a subset of individuals from a population to estimate characteristics of the whole group. In our case, we have a scenario where a sample of ten 18-20 year olds is taken from a broader population for analysis.

When creating a sample, it's essential to ensure that it is representative of the entire population. Random sampling can achieve this by giving every individual an equal chance of being picked. This prevents bias and helps ensure accurate statistical inferences.

The size of the sample, as well as the randomness of selection, directly impact the reliability of the results you can expect from calculations like those in binomial distribution problems. When you're working with smaller samples, like in this scenario, remember that variability increases, affecting confidence in the inferences drawn.

Independent Trials

In statistics, independent trials mean the outcome of one trial does not affect the outcome of another. This is fundamental to the binomial distribution, where each trial must be independent to appropriately use the distribution.

For our underage drinking example, each 18-20 year old's decision to consume an alcoholic beverage is independent of the others in the sample. This means the behavior of one individual does not influence another's decision in the context of the sampling.

The guarantee of independent trials ensures that the calculated probabilities reflect the reality as closely as possible. If trials were not independent, we'd be unable to rely on the calculated probabilities, as they would not account for the interconnectedness of events.

Success and Failure Outcomes

Success and failure outcomes are critical concepts in binomial probability. Here, every trial results in either a 'success' or a 'failure'. For underage drinking, a 'success' might be defined as an individual consuming an alcoholic beverage, and a 'failure' would be the opposite.

These outcomes are binary, meaning there are only two possible outcomes. Assigning these helps when performing probability calculations because it simplifies complex real-world events into manageable statistical events. The probability of success and failure remains constant across trials.

Understanding what qualifies as success or failure lets you apply the binomial distribution formula accurately. Matching real-world scenarios to these outcomes provides clarity and allows for feasible computational efforts in analysis.