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In the Healthy Hand washing Survey conducted by Bradley Corporation, it was found that \(64 \%\) of adult Americans operate the flusher of toilets in public restrooms with their foot. Suppose a random sample of \(n=20\) adult Americans is obtained and the number \(x\) who flush public toilets with their foot is recorded. (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 12 flush public toilets with their foot. (c) Find and interpret the probability that at least 16 flush public toilets with their foot. (d) Find and interpret the probability that between 9 and 11 , inclusive, flush public toilets with their foot. (e) Would it be unusual to find more than 17 who flush public toilets with their foot? Why?

Step by step solution

01

Explain why this is a binomial experiment (Part a)

A binomial experiment has four characteristics: (1) the experiment consists of a fixed number of trials, (2) each trial is independent, (3) there are only two possible outcomes for each trial (success or failure), and (4) the probability of success is the same for each trial. In this case, there are 20 trials (each adult sampled), each trial (each adult's response) is independent, there are two possible outcomes (flush with foot or not), and the probability of success (flush with foot) is 0.64. Therefore, this is a binomial experiment.

02

Identify binomial variables (Part b, c, d, e)

Denote the probability of success (flushing with foot) as \(p = 0.64\) and the number of trials as \(n = 20\). The random variable \(X\), which represents the number of people who flush with their foot, follows a binomial distribution: \(X \sim B(n=20, p=0.64)\). The probability mass function of a binomial distribution is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

03

Calculate the probability that exactly 12 flush public toilets with their foot (Part b)

Using the binomial probability formula,\[ P(X = 12) = \binom{20}{12} (0.64)^{12} (1 - 0.64)^{8} \]First calculate \(\binom{20}{12}\), then raise 0.64 to the 12th power and (1-0.64) to the 8th power, and finally multiply the results together. The probability is approximately 0.114.

04

Interpret the probability that exactly 12 flush public toilets with their foot (Part b)

The probability that exactly 12 out of 20 sampled adults flush public toilets with their foot is about 0.114 or 11.4%, meaning it is somewhat likely to observe exactly 12 adults who flush with their foot in the sample.

05

Calculate the probability that at least 16 flush public toilets with their foot (Part c)

To find the probability that at least 16 adults flush public toilets with their foot, sum the probabilities for 16, 17, 18, 19, and 20.\[ P(X \geq 16) = P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) \]Calculate each probability using the binomial formula and add them together. The result is approximately 0.022.

06

Interpret the probability that at least 16 flush public toilets with their foot (Part c)

The probability that at least 16 adults out of 20 flush public toilets with their foot is about 0.022 or 2.2%, indicating it is quite unlikely to observe 16 or more adults who flush with their foot in the sample.

07

Calculate the probability that between 9 and 11, inclusive, flush public toilets with their foot (Part d)

Sum the probabilities of having exactly 9, 10, and 11 adults who flush with their foot.\[ P(9 \leq X \leq 11) = P(X = 9) + P(X = 10) + P(X = 11) \]Calculate each probability using the binomial formula and add them together. The result is approximately 0.258.

08

Interpret the probability that between 9 and 11, inclusive, flush public toilets with their foot (Part d)

The probability that between 9 and 11 adults (inclusive) flush public toilets with their foot is about 0.258 or 25.8%, meaning it is fairly common to observe this range in the sample.

09

Determine if it is unusual to find more than 17 who flush public toilets with their foot (Part e)

An event is considered unusual if its probability is less than 0.05. Find the probability that more than 17 adults flush with their foot:\[ P(X > 17) = P(X = 18) + P(X = 19) + P(X = 20) \]Calculate each probability using the binomial formula and add them together. The result is approximately 0.004.

10

Interpret the results for more than 17 flushing with their foot (Part e)

The probability that more than 17 adults out of 20 flush public toilets with their foot is about 0.004 or 0.4%, indicating it is very unusual to find more than 17 who flush with their foot in the sample.

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Key Concepts

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

Binomial Distribution

In statistics, the binomial distribution is a key concept. It's used to model the number of successes in a fixed number of binary (yes/no) trials. Here, we've identified our trials as asking 20 adults if they flush public toilets with their foot. Each response is independent, meaning the answer from one adult doesn't influence another's response. We only have two outcomes: either they do flush with their foot (success) or they don't (failure). Thus, our scenarios fit into this distribution perfectly with probabilities consistent across trials.

Probability Calculation

Calculating probabilities in a binomial experiment involves understanding the binomial probability formula. For example:
\(
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\)

This formula tells us the probability of exactly k successes out of n trials, where p is the probability of success on each trial. For instance, to find the likelihood of exactly 12 out of 20 adults flushing with their foot, we'd set k=12, n=20, and p=0.64. Plugging these values into our formula gives us an accurate probability value, helping us to understand the likelihood of any given scenario.

Binomial Probability Formula

The binomial probability formula is a crucial tool for calculating specific probabilities in binomial distributions. It combines combinations (n choose k) with the probabilities of success and failure.

For our example, we calculate combinations with:

\(\binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Next, multiply this by the probability of success raised to the power of successes, and the probability of failure raised to the power of failures. This formula, while looking complex at first, becomes straightforward with practice and is essential for accurate binomial probability calculations.

Statistical Analysis

Utilizing statistical analysis in binomial experiments allows us to interpret our probability calculations meaningfully. For example, knowing that the probability of at least 16 adults (out of 20) flushing with their foot is 2.2% indicates a rare event. Similarly, probabilities between specific ranges help in understanding common outcomes. If we find that between 9 and 11 adults doing so is about 25.8%, this frequency suggests a common occurrence. Statistical analysis involves estimating parameters, testing hypotheses, and using p-values to make data-based decisions regarding the likelihood and frequency of events in a binomial experiment.