Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.T.5b

Chapter 4, Problem 4.T.5b

The table shows the ages of students in a freshman orientation course.

b. Graph the probability distribution using a histogram and describe its shape.

Verified step by step guidance

Step 1: Calculate the probability distribution. Divide the number of students for each age by the total number of students to find the probability for each age. For example, if there are 25 students in total, the probability for age 17 would be $ P(17) = \frac{2}{25} $. Repeat this for all ages.

Step 2: Create a histogram. On the x-axis, plot the ages (17, 18, 19, 20, 21, 22). On the y-axis, plot the probabilities calculated in Step 1. Each bar's height will correspond to the probability of the respective age.

Step 3: Label the histogram. Ensure the x-axis is labeled as 'Age' and the y-axis is labeled as 'Probability'. Add a title such as 'Probability Distribution of Student Ages'.

Step 4: Analyze the shape of the histogram. Observe whether the distribution is symmetric, skewed left, skewed right, or uniform. For example, if most probabilities are concentrated around a central age, the distribution might be symmetric.

Step 5: Summarize the shape. Based on the histogram, describe the overall pattern. For instance, if the probabilities decrease as age increases, the distribution might be skewed to the right.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. In this context, it represents the likelihood of each age occurring among the students in the orientation course. The distribution can be visualized using a histogram, where the x-axis represents the ages and the y-axis represents the frequency of students at each age.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where data is grouped into bins or intervals. Each bar's height corresponds to the frequency of data points within that interval. In this case, the histogram will display the ages of students on the x-axis and the number of students on the y-axis, allowing for a visual interpretation of the age distribution.

Shape of Distribution

The shape of a distribution refers to the visual appearance of the histogram, which can indicate patterns such as symmetry, skewness, or modality. For example, a bell-shaped curve suggests a normal distribution, while a right or left skew indicates that more data points are concentrated on one side. Analyzing the shape helps in understanding the underlying characteristics of the data, such as age trends among students.

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Related Practice

Textbook Question

In Exercises 1–3, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (b) the first return requiring an audit is the first or second return the tax auditor examines,

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

b. more than four customers will arrive during the first minute.

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Textbook Question

One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (a) the first return requiring an audit is the 25th return the tax auditor examines, (b) the first return requiring an audit is the first or second return the tax auditor examines, and (c) none of the first five returns the tax auditor examines require an audit. (Source: Kiplinger)

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

a. three, four, or five customers will arrive during the third minute.

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Textbook Question

One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (c) none of the first five returns the tax auditor examines require an audit.

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

c. more than four customers will arrive during each of the first four minutes.

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The table shows the ages of students in a freshman orientation course.b. Graph the probability distribution using a histogram and describe its shape.

Verified video answer for a similar problem:

Key Concepts

Probability Distribution

Histogram

Shape of Distribution

The table shows the ages of students in a freshman orientation course.

b. Graph the probability distribution using a histogram and describe its shape.