91Ó°ÊÓ

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 58 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 Criteria

The binomial distribution requires three conditions: a fixed number of trials, only two outcomes (success/failure), and the probability of success remains constant. Here, we have 10 trials (18-20 year olds), two outcomes (consumed or not), and a consistent probability of consuming being 69.7%. The binomial distribution is appropriate.

02

Define Probability of Success

Success is defined as a student consuming alcoholic beverages. Let the probability of success, \( p = 0.697 \). The probability of failure, \( q = 1 - p = 0.303 \).

03

Calculate Probability of Exactly 6 Drinkers in 10

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \). For \( n = 10 \), \( k = 6 \), \( p = 0.697 \): \[ P(X = 6) = \binom{10}{6} (0.697)^6 (0.303)^4 \]. Solve to find the probability.

04

Calculate Probability of Exactly 4 Non-Drinkers in 10

If exactly 4 out of 10 haven't consumed, then exactly 6 have consumed. Therefore, use the result from Step 3, \( P(X = 6) \), since the event is the same under rephrased conditions.

05

Calculate Probability of At Most 2 Drinkers in 5

For \( n = 5 \), this is the sum of probabilities \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \): \[ P(X \leq 2) = \binom{5}{0}(0.697)^0(0.303)^5 + \binom{5}{1}(0.697)^1(0.303)^4 + \binom{5}{2}(0.697)^2(0.303)^3 \]. Calculate each term and sum them.

06

Calculate Probability of At Least 1 Drinker in 5

The complement rule helps here: \[ P(X \geq 1) = 1 - P(X = 0) \]. Using \( n = 5 \): \[ P(X = 0) = \binom{5}{0}(0.697)^0(0.303)^5 \]. Subtract this from 1 to find \( P(X \geq 1) \).

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Key Concepts

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

Probability Calculations

Probability calculations guide us in determining how likely an event is to occur. It's a branch of mathematics that deals with finding numerical values for the likelihood of specific outcomes.
In the context of the exercise, calculating the probability involves determining how likely it is that a certain number of individuals in a sample behaved in a specific way, such as consuming alcoholic beverages.

• The probability scale ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
• Calculations involve a clear understanding of the event being analyzed, the total number of possible outcomes, and the number of desired outcomes.
• For instance, if we want to find the probability of exactly six individuals consuming alcohol out of ten, we would first identify the probability of success (consuming alcohol) and failure (not consuming).

By applying formulas and rules, like the binomial probability which we’ll discuss next, you can make accurate predictions and understand trends within the data.

Binomial Probability Formula

The binomial probability formula is a critical tool in statistics for finding the probability of a given number of successes in a specific number of trials. In our context, the trials involve each individual encountered in the sample.
The formula is given by:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \]

• \( n \) is the total number of trials, which is the number of individuals sampled.
• \( k \) is the number of successes, such as the number of people who consumed alcohol.
• \( p \) is the probability of success on an individual trial.
• The term \( \binom{n}{k} \) represents the number of combinations for selecting \( k \) successes out of \( n \) possible trials.

The formula tells us how to organize these values to calculate the likelihood of a specified event. In our case, it helps in calculating the probability for different scenarios like exactly six out of ten consuming alcohol or finding the likelihood of non-consumers.

Complement Rule

The complement rule is a clever trick in probability that simplifies finding probabilities with a yes-or-no question. It states that the probability of event \( A \) happening is 1 minus the probability of it not happening.
This can be expressed as:
\[ P(A) = 1 - P(A') \]
where \( P(A') \) is the probability of the event not occurring.

• If finding the probability of an event is complex, consider using its complement. For example, to find the probability of at least one person consuming alcohol, it's often easier to calculate the probability that none did and subtract from 1.
• This rule is particularly useful in our exercise when calculating the probability of at least one drinker in a sample of five, which derives from finding no drinkers’ probability and subtracting from 1.

Utilizing the complement rule can save time and calculations when dealing with multiple or complex probability scenarios.

Sampling in Statistics

Sampling in statistics helps researchers make inferences about a larger population based on a small, manageable group. This smaller group is called a sample, and it should accurately represent the population to make valid conclusions.
In our exercise, the sample consists of ten random 18-20 year olds, which represents the larger group of all 18-20 year olds in the given year.

• Random sampling ensures unbiased data, as every individual in the population has an equal chance of being chosen.
• A representative sample allows statisticians to apply probability distributions, such as the binomial distribution, to determine the likelihood of various outcomes.
• Using a proper sampling technique means conclusions about the overall habits, like alcohol consumption, are more reliable.

Understanding how sampling works and ensuring proper sampling techniques are vital steps in conducting accurate statistical research and drawing valid conclusions about broader populations.