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Hiking the High Sierra loop in Yosemite. Yosemite National Park has five High Sierra camps arranged in a 49-mile loop. The camps are separated by 8-10 miles, which allows you to hike from one camp to the next in a day. Each camp provides tent cabins with four to six beds per cabin and serves family-style meals. Because of the popularity of the loop, you must enter a lottery in the fall for the following summer, and in a given year, approximately \(25 \%\) of the groups are selected at random to hike the loop." You plan to enter the lottery each year for the next five years. Let \(X\) be the number of years in which you are selected to hike the loop. (a) \(X\) has a binomial distribution. What are \(n\) and \(p\) ? (b) What are the possible values that \(X\) can take? (c) Find the probability of each value of \(X\). Draw a probability histogram for the distribution of \(\boldsymbol{X}\). (See Figure 14.2, page 339, for an example of a probability histogram-) (d) What are the mean and standard deviation of this distribution? Mark the location of the mean on your histogram.

Step by step solution

01

Identify the Parameters

For a binomial distribution, two parameters are crucial: \( n \), which represents the number of trials, and \( p \), the probability of success in each trial. Here, \( n = 5 \) (hiking lottery entered each year for 5 years), and \( p = 0.25 \) (the probability of being selected in any given year as per the lottery rule).

02

Determine Possible Values of X

The random variable \( X \), which represents the number of successful selections over the 5 years, can take any integer value from 0 to 5. Thus, the possible values of \( X \) are \( \{0, 1, 2, 3, 4, 5\} \).

03

Calculate Probabilities

For a binomial distribution, the probability of getting exactly \( k \) successes (or selected years) is given by the formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). Calculate these probabilities for each \( k \) from 0 to 5 using \( n = 5 \) and \( p = 0.25 \).

04

Construct a Probability Histogram

Create a histogram using the calculated probabilities from Step 3. Each bar on the histogram corresponds to a possible value of \( X \), with the height representing the probability of that value.

05

Compute the Mean and Standard Deviation

The mean of the binomial distribution is given by \( \mu = np \) and the standard deviation by \( \sigma = \sqrt{np(1-p)} \). Calculate these using \( n = 5 \) and \( p = 0.25 \). Plot the mean on the histogram.

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Key Concepts

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

Probability

Probability is a measure of how likely an event is to occur. In the context of our hiking lottery problem, it helps us understand the likelihood of being selected to hike the High Sierra loop in Yosemite over a span of years. When dealing with probability, events are usually given a number between 0 and 1, where 0 means the event will not happen and 1 means it will definitely happen.
In our exercise, we know the probability of being selected in any given year is 0.25 or 25%. This probability value is fundamental to solving the problem using a binomial distribution.

• **Example**: If you roll a fair six-sided die, the probability of rolling a "3" is \( \frac{1}{6} \).
• **Key Point**: Probability helps us make predictions about how often random events will occur.

Statistics

Statistics is the field that deals with collecting, analyzing, interpreting, and presenting data. It helps us summarize complex datasets and uncover patterns and insights from them. In our problem, understanding statistics involves recognizing that the number of successful lottery entries over five years can be modeled using statistical methods like the binomial distribution.
By employing statistical tools, such as histograms, we can visualize these probabilities effectively. A histogram shows how often each outcome happens, making it easier to understand and interpret the data.

• **Example**: If we collect ages of people in a room and summarize them to find the average, that's statistics in action.
• **Key Point**: Statistics allows us to deal with variability in data and make informed decisions based on it.

Random Variable

A random variable is a variable used in probability and statistics to quantify outcomes of random phenomena. It assigns numerical values to each outcome in a probability space. In our exercise, the random variable \( X \) represents the number of years you get selected to hike.
Random variables can be either discrete (finite possible values) or continuous (infinite possible values). Since \( X \) can only take on whole numbers (from 0 to 5 in this case), it is a discrete random variable.

• **Example**: Flipping a coin three times and counting the number of heads is a discrete random variable.
• **Key Point**: Discrete random variables like \( X \) help us model scenarios with finite outcomes, allowing us to apply probability calculations.

Mean and Standard Deviation

The mean and standard deviation are essential statistical concepts that describe a dataset. The mean is the average value, providing a central location for the data. In the context of a binomial distribution, the mean gives us the expected number of successful outcomes. For our exercise, the mean \( \mu = np \) shows the average number of times you can expect to be selected over five years, which comes out to \( 5 \times 0.25 = 1.25 \).
The standard deviation, on the other hand, measures the dispersion or spread of the data around the mean. For a binomial distribution, it's calculated by the formula \( \sigma = \sqrt{np(1-p)} \). This shows us the typical amount that returned values deviate from the mean.

• **Example**: If you measure the heights of a group of people, the mean height is the average, and the standard deviation quantifies how varied the heights are.
• **Key Point**: Mean provides a central tendency, while standard deviation indicates variability within the data.