For the binomial distribution Upper P left parenthesis x right parenthesis equals Upper C Subscript 6 comma x Baseline left parenthesis 0.3 right parenthesis Superscript x Baseline left parenthesis 0.7 right parenthesis Superscript 6 minus x​, choose the correct histogram below and compute the mean and standard deviation for the distribution.
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Part 1
Choose the correct histogram below.
A.
A bar graph has a horizontal axis and a vertical axis labeled from 0 to 0.5 in increments of 0.1. 7 vertical bars extend up above the horizontal axis. From left to right, the heights of the bars are as follows: 0.118, 0.303, 0.324, 0.185, 0.06, 0.01, 0.0007. All heights are approximate.
0
0.1
0.2
0.3
0.4
0.5
P(x)
B.
A bar graph has a horizontal axis and a vertical axis labeled from 0 to 0.5 in increments of 0.1. 6 vertical bars extend up above the horizontal axis. From left to right, the heights of the bars are as follows: 0.01, 0.06, 0.185, 0.185, 0.303, 0.118. All heights are approximate.
0
0.1
0.2
0.3
0.4
0.5
P(x)
C.
A bar graph has a horizontal axis and a vertical axis labeled from 0 to 0.5 in increments of 0.1. 8 vertical bars extend up above the horizontal axis. From left to right, the heights of the bars are as follows: 0.118, 0.303, 0.324, 0.185, 0.185, 0.324, 0.303, 0.118. All heights are approximate.
0
0.1
0.2
0.3
0.4
0.5
P(x)
D.
A bar graph has a horizontal axis and a vertical axis labeled from 0 to 0.5 in increments of 0.1. 8 vertical bars extend up above the horizontal axis. From left to right, the heights of the bars are as follows: 0.185, 0.324, 0.303, 0.118, 0.118, 0.303, 0.324, 0.185. All heights are approximate.
0
0.1
0.2
0.3
0.4
0.5
P(x)

To solve this problem, we first need to understand the binomial distribution and its properties.

### Binomial Distribution
The binomial distribution \( P(x) \) for \( n \) trials and success probability \( p \) is given by:

\[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \]

Here, \( n = 6 \) and \( p = 0.3 \). So the binomial probability is:

\[ P(x) = \binom{6}{x} (0.3)^x (0.7)^{6-x} \]

Let's calculate the probabilities for \( x = 0 \) to \( x = 6 \):

1. \( P(0) = \binom{6}{0} (0.3)^0 (0.7)^6 \approx 0.118 \)
2. \( P(1) = \binom{6}{1} (0.3)^1 (0.7)^5 \approx 0.303 \)
3. \( P(2) = \binom{6}{2} (0.3)^2 (0.7)^4 \approx 0.324 \)
4. \( P(3) = \binom{6}{3} (0.3)^3 (0.7)^3 \approx 0.185 \)
5. \( P(4) = \binom{6}{4} (0.3)^4 (0.7)^2 \approx 0.060 \)
6. \( P(5) = \binom{6}{5} (0.3)^5 (0.7)^1 \approx 0.010 \)
7. \( P(6) = \binom{6}{6} (0.3)^6 (0.7)^0 \approx 0.0007 \)

Now we compare these values to the provided options:

- Option A matches our calculated probabilities:
  - \( 0.118, 0.303, 0.324, 0.185, 0.060, 0.010, 0.0007 \)

- Option B, C, and D have different values or numbers of bars.

Thus, the correct histogram is:

### Option A
A bar graph with 7 vertical bars extending up above the horizontal axis. From left to right, the heights of the bars are as follows: 0.118, 0.303, 0.324, 0.185, 0.06, 0.01, 0.0007. All heights are approximate.

### Mean and Standard Deviation

For a binomial distribution:
- The mean \( \mu \) is given by: 
  \[ \mu = n \cdot p \]
- The standard deviation \( \sigma \) is given by: 
  \[ \sigma = \sqrt{n \cdot p \cdot (1-p)} \]

Plugging in the values \( n = 6 \) and \( p = 0.3 \):

- Mean:
  \[ \mu = 6 \cdot 0.3 = 1.8 \]

- Standard Deviation:
  \[ \sigma = \sqrt{6 \cdot 0.3 \cdot 0.7} = \sqrt{1.26} \approx 1.122 \]

### Summary
- **Correct Histogram:** Option A
- **Mean:** 1.8
- **Standard Deviation:** 1.122

Would you like more details on any specific part of the solution? 

### Additional Questions
1. How do you calculate the binomial coefficient \( \binom{n}{x} \)?
2. Why does the binomial distribution use \( (1-p) \) in the formula?
3. What are the applications of the binomial distribution in real-world scenarios?
4. How does changing \( p \) affect the shape of the binomial distribution?
5. Can you derive the mean and standard deviation formulas for the binomial distribution?
6. What is the significance of the probabilities summing to 1 in a binomial distribution?
7. How would you plot a binomial distribution histogram for a different \( n \) and \( p \)?
8. How can we use the binomial distribution to approximate other distributions?

### Tip
When dealing with binomial distributions, remember that they model the number of successes in a fixed number of independent trials, each with the same probability of success. This can be helpful in understanding why the formulas and results behave as they do.

Explore the binomial distribution with a focus on calculating mean, standard deviation, and analyzing histograms. Understand the formula P(x) = C(6, x) * 0.3^x * 0.7^(6-x) for a binomial experiment with n=6 trials and success probability p=0.3. Find the correct histogram representation and compute statistical measures.

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