P(x) .40 0.20971520 1 0.36700160 P(x) 30 0.27525120 0.11468800 .20 4 0.02867200 0.00430080 .10 6. 0.00035840 7. 0.00001280 0 1 2 3 4 5 6 7
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.
![## Educational Content: Understanding Binomial Distribution
**Binomial Distribution Overview:**
- **X ~ Binomial (n = __, p = __):** This represents a binomial distribution where `n` is the number of trials and `p` is the probability of success on each trial.
### Probability Distribution Table
For the random variable \( X \), representing the number of adults with no trouble sleeping, the probabilities are as follows:
| x | P(x) |
|----|--------------|
| 0 | 0.20971520 |
| 1 | 0.36700160 |
| 2 | 0.27525120 |
| 3 | 0.114668800 |
| 4 | 0.028267200 |
| 5 | 0.004300800 |
| 6 | 0.00035840 |
| 7 | 0.00001280 |
### Probability Histogram Explanation
- The histogram is a graphical depiction showing the probability \( P(x) \) on the y-axis against the number of adults with no trouble sleeping (x) on the x-axis. It represents discrete probabilities for each value of \( X \) from 0 to 7, reflecting how likely each scenario is.
### Questions and Calculations
c. **Probability that at least one has no trouble sleeping:**
\[
P(X \geq 1) = 1 - P(X = 0)
\]
d. **Probability that at most 2 have no trouble sleeping:**
\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]
e. **Probability that more than 4 have no trouble sleeping:**
\[
P(X > 4) = P(X = 5) + P(X = 6) + P(X = 7)
\]
f. **Find \( \mu_X \) and \( \sigma_X \) for this distribution:**
- **Mean (\( \mu_X \)):** The average expected number of successes.
- **Standard Deviation (\( \sigma_X \)):** Measures the dispersion of the distribution.
**Interpretation:**
- \( \mu_X \) provides information on the expected average](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1064d08b-e094-4e7d-bb6f-10908ee8847f%2F23507ff3-51ba-4a5a-b5c7-a51b0f0f485f%2Fxr8l5bk_processed.jpeg&w=3840&q=75)
[Note: Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.]
A probability histogram is a graph which shows the probability of each outcome on the y-axis and outcomes in the x-axis.
Here x denotes the outcomes and p(x) denotes the probabilities.
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