2. Binomial Experiment: A sleep study determines that one in five adults say they have no trouble sleeping at night. You randomly select seven adults and ask each if they have no trouble sleeping. Let X be the number of people who have no trouble sleeping at night. Determine if this is a binomial problem. a. 1. 2. 3. X~Binomial( n= p = b. Find the probability distribution table and draw the probability histogram for X. 40 P(x) 0. 0.20971520 0.36700160 P(x) 30 1 0.27525120 20 3. 0.11468800 4 0.02867200 .10 0.00430080 6. 0.00035840 0.00001280 0 1 2 3 4 5 6 7 7. c. Find the probability that at least one has no trouble sleeping. P( X )= d. Find the probability that at most 2 have no trouble sleeping. P( X ): e. Find the probability that more than 4 have no trouble sleeping. P(X _) ES<2> f. Find u and a for this distribution mean u st. dev a=

2. Binomial Experiment: A sleep study determines that one in five adults say they have no trouble sleeping at night. You randomly select seven adults and ask each if they have no trouble sleeping. Let X be the number of people who have no trouble sleeping at night. Determine if this is a binomial problem. a. 1. 2. 3. X~Binomial( n= p = b. Find the probability distribution table and draw the probability histogram for X. 40 P(x) 0. 0.20971520 0.36700160 P(x) 30 1 0.27525120 20 3. 0.11468800 4 0.02867200 .10 0.00430080 6. 0.00035840 0.00001280 0 1 2 3 4 5 6 7 7. c. Find the probability that at least one has no trouble sleeping. P( X )= d. Find the probability that at most 2 have no trouble sleeping. P( X ): e. Find the probability that more than 4 have no trouble sleeping. P(X _) ES<2> f. Find u and a for this distribution mean u st. dev a=

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Description: The document details a section on understanding binomial distributions with specific reference to a probability distribution table and histogram representation.**Binomial Distribution Example:**- X ~ Binomial(n = ___, p = ___)**Tasks:**b. **Probabil

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

The document details a section on understanding binomial distributions with specific reference to a probability distribution table and histogram representation.

**Binomial Distribution Example:**

- X ~ Binomial(n = ___, p = ___)

**Tasks:**

b. **Probability Distribution Table and Histogram for X:**

| x | P(x) |
|---|---------------|
| 0 | 0.20971520 |
| 1 | 0.36700160 |
| 2 | 0.27525120 |
| 3 | 0.11468800 |
| 4 | 0.02667200 |
| 5 | 0.00343080 |
| 6 | 0.00035840 |
| 7 | 0.00001280 |

- *Histogram*: The graph presents a histogram with P(x) on the y-axis ranging from 0 to 40, and x-values from 0 to 7. Each bar represents the probability P(x) for each respective x-value.

c. **Probability Calculations:**
- Find the probability that at least one has no trouble sleeping. $P(X \geq 1) =$

d. - Find the probability that at most 2 have no trouble sleeping. $P(X \leq 2) =$

e. - Find the probability that more than 4 have no trouble sleeping. $P(X > 4) =$

f. **Statistical Measures:**
- Find $ \mu_x $ and $ \sigma_x $ for this distribution.
- Mean $ \mu_x = $ ______
- Standard deviation $ \sigma_x = $ ______

**Interpretation:**
- $ \mu_x $ indicates that for the distribution, if seven random adults were selected repeatedly, on average, we would expect _____ out of every seven to have the counted characteristic.

# Binomial Experiment

A sleep study determines that one in five adults say they have no trouble sleeping at night. You randomly select seven adults and ask each if they have no trouble sleeping.

### a. Determine if this is a Binomial Problem

Let $ X $ be the number of people who have no trouble sleeping at night.

1.
2.
3.
4.

\[ X = \text{Binomial}(n = \_\_\_\_\_, p = \_\_\_\_\_) \]

### b. Constructing the Probability Distribution Table and Histogram

**Probability Distribution Table**

| $ x $ | $ P(x) $ |
|---------|-----------------|
| 0 | 0.20971520 |
| 1 | 0.36700160 |
| 2 | 0.27525120 |
| 3 | 0.11468800 |
| 4 | 0.02867200 |
| 5 | 0.00430080 |
| 6 | 0.00035840 |
| 7 | 0.00001280 |

**Probability Histogram**

- The histogram is a graphical representation of the probability distribution.
- The x-axis represents the number of adults with no trouble sleeping (0 to 7).
- The y-axis represents the probabilities $ P(x) $.
- Bars are plotted corresponding to the probabilities in the table. The height of each bar equals the probability for that number of people having no trouble sleeping.

### c. Probability Calculations

1. **Probability that at least one has no trouble sleeping.**

$ P(X \geq 1) = \_\_\_\_\_ $

2. **Probability that at most 2 have no trouble sleeping.**

$ P(X \leq 2) = \_\_\_\_\_ $

3. **Probability that more than 4 have no trouble sleeping.**

$ P(X > 4) = \_\_\_\_\_ $

### f. Calculating Mean and Standard Deviation

- **Mean ($ \mu $):** \_\_\_\_\_
- **Standard Deviation ($ \sigma $):** \_\_\_\_\_

This content is presented for educational purposes to illustrate the process of analyzing binomial probabilities and interpreting statistical data through calculated probabilities

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