Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Statistics

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

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2:13 minutes

Problem 4.1.25

Textbook Question

Determining a Missing Probability In Exercises 25 and 26, determine the missing probability for the probability distribution.

Verified step by step guidance

Step 1: Recall that for a probability distribution, the sum of all probabilities must equal 1. This is a fundamental property of probability distributions.

Step 2: Write the equation to represent the sum of probabilities: \( P(0) + P(1) + P(2) + P(3) + P(4) = 1 \). Substitute the known values: \( 0.06 + 0.12 + 0.18 + P(3) + 0.30 = 1 \).

Step 3: Combine the known probabilities: \( 0.06 + 0.12 + 0.18 + 0.30 = 0.66 \). This simplifies the equation to \( 0.66 + P(3) = 1 \).

Step 4: Solve for \( P(3) \) by subtracting \( 0.66 \) from both sides: \( P(3) = 1 - 0.66 \).

Step 5: Verify that the sum of all probabilities equals 1 after finding \( P(3) \). This ensures the distribution is valid.

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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. For discrete random variables, it lists each possible value and its corresponding probability, ensuring that the sum of all probabilities equals 1. Understanding this concept is crucial for determining any missing probabilities in a given distribution.

Sum of Probabilities

In a valid probability distribution, the sum of all probabilities must equal 1. This principle is essential for solving problems involving missing probabilities, as it allows us to set up an equation where the unknown probability can be calculated by subtracting the sum of known probabilities from 1.

Complementary Events

Complementary events are pairs of outcomes in a probability space that cover all possible outcomes. In the context of probability distributions, if we know the probabilities of certain outcomes, we can find the probability of the missing outcome by recognizing that it is the complement of the sum of the known probabilities. This concept aids in efficiently determining missing values.

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Textbook Question

Graphical Analysis In Exercises 9–12, determine whether the graph on the number line represents a discrete random variable or a continuous random variable. Explain your reasoning.

The distance a baseball travels after being hit

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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.

Minitab was used to generate 20 random numbers with a Poisson distribution for . Let the random number represent the number of arrivals at the checkout counter each minute for 20 minutes. 3 3 3 3 5 5 6 7 3 6 3 5 6 3 4 6 2 2 4 1During each of the first four minutes, only three customers arrived. These customers could all be processed, so there were no customers waiting after four minutes.

b. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes.

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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 increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.

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Textbook Question

Unusual Events In Exercise 19, would it be unusual for a household to have no HD televisions? Explain your reasoning.

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Textbook Question

Determining a Missing Probability In Exercises 25 and 26, determine the missing probability for the probability distribution.

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Textbook Question

Identifying Probability Distributions In Exercises 27 and 28, determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

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Textbook Question

"True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

The mean of the random variable of a probability distribution describes how the outcomes vary."

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Multiple Choice

A student is analyzing different types of variables in a statistics class. Which of the following below is a discrete random variable?