Theoretical DistributionUsing the theoretical distribution, complete the following statements. Use a normalapproximation based on the sample mean and standard deviation.IQR = _______The 15 percentile is _______.The 85 percentile is _______.Median is _______. Attached I have the data I am currently working with and the assignment. I need help with the Theoretical Distribution section above. ### Calculate the Following3. Calculate the following. - a. \( \bar{x} = \_\_\_\_\_ \) - b. \( s = \_\_\_\_\_ \)4. Draw a smooth curve through the top of the bars of the histogram. Write one to two complete sentences to describe the general shape of the curve. (Keep it simple. Does the graph go straight across, does it have a v-shape, does it have a hump in the middle or at either end, and so on?)### Analyze the DistributionUsing your sample mean, sample standard deviation, and histogram, what was the approximate theoretical distribution of the data you collected?- \( X \sim \_\_\_ \left( \_\_\_\_\_ , \_\_\_\_\_\_ \right) \)- How does the histogram help you arrive at the approximate distribution?### Describe the DataUsing the data you collected complete the following statements. (Hint: order the data)#### Remember\( \text{IQR} = Q_3 - Q_1 \)- **IQR = \_\_\_\_\_**- The **15th percentile** is \_\_\_\_\_.- The **85th percentile** is \_\_\_\_\_.- **Median** is \_\_\_\_\_.- What is the theoretical probability that a randomly chosen pinky length is more than 6.5 cm?- Explain the meaning of the **85th percentile** of this data.### Theoretical DistributionUsing the theoretical distribution, complete the following statements. Use a normal approximation based on the sample mean and standard deviation.- **IQR = \_\_\_\_\_**- The **15th percentile** is \_\_\_\_\_.- The **85th percentile** is \_\_\_\_\_.- **Median** is \_\_\_\_\_. ### Data on Pinky Length#### Table of Pinky Length Measurements- Measurements: 7, 7, 7, 7, 6.5, 6.5, 6.5, 6.5, 6, 6, 7, 7, 5, 6.5, 7, 6.5, 6, 6.5, 6.5, 6, 5.5, 6.5, 6, 6.5, 6.5, 6, 6, 5.5, 6, 6.5#### Statistical Summary- **Average (Mean):** 6.433333- **Standard Deviation (S):** 0.486602- **1st Quartile (Q1):** 6- **2nd Quartile (Q2/Median):** 6.5- **Interquartile Range (IQR):** 0.875- **15th Percentile:** 6- **85th Percentile:** 7- **Normal Distribution Approximation:** - \( X \sim \mathcal{N}(6.43, 0.487) \) #### Probability Calculation- **Probability of Length Greater than 6.5 cm:** - \( P(X > 6.5) = 0.266667 \)#### Histogram DescriptionThe histogram visualizes the frequency distribution of pinky lengths:- **Intervals:** 1. [5, 5.55] 2. (5.55, 6.1] 3. (6.1, 6.65] 4. (6.65, 7.2] - **Frequency:** - [5, 5.55]: 2 - (5.55, 6.1]: 6 - (6.1, 6.65]: 13 (most frequent) - (6.65, 7.2]: 9This data and its analysis allow us to understand the distribution of pinky lengths in the sample, highlighting the central tendency, variability, and the probability of certain lengths.

Given that: Pinky Length: 7, 7, 7, 6.5, 6.5, 6.5, 6.5, 6, 7, 7, 7, 5, 6, 7, 6.5, 6.5, 6, 6.5, 6,…

Answered: Theoretical Distribution Using the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Theoretical Distribution
Using the theoretical distribution, complete the following statements. Use a normal
approximation based on the sample mean and standard deviation.
IQR = _______
The 15 percentile is _______.
The 85 percentile is _______.
Median is _______.

Attached I have the data I am currently working with and the assignment. I need help with the Theoretical Distribution section above.

### Calculate the Following

3. Calculate the following.
- a. \( \bar{x} = \_\_\_\_\_ \)
- b. \( s = \_\_\_\_\_ \)

4. Draw a smooth curve through the top of the bars of the histogram. Write one to two complete sentences to describe the general shape of the curve. (Keep it simple. Does the graph go straight across, does it have a v-shape, does it have a hump in the middle or at either end, and so on?)

### Analyze the Distribution

Using your sample mean, sample standard deviation, and histogram, what was the approximate theoretical distribution of the data you collected?
- \( X \sim \_\_\_ \left( \_\_\_\_\_ , \_\_\_\_\_\_ \right) \)
- How does the histogram help you arrive at the approximate distribution?

### Describe the Data

Using the data you collected complete the following statements. (Hint: order the data)

#### Remember
\( \text{IQR} = Q_3 - Q_1 \)

- **IQR = \_\_\_\_\_**
- The **15th percentile** is \_\_\_\_\_.
- The **85th percentile** is \_\_\_\_\_.
- **Median** is \_\_\_\_\_.
- What is the theoretical probability that a randomly chosen pinky length is more than 6.5 cm?
- Explain the meaning of the **85th percentile** of this data.

### Theoretical Distribution

Using the theoretical distribution, complete the following statements. Use a normal approximation based on the sample mean and standard deviation.

- **IQR = \_\_\_\_\_**
- The **15th percentile** is \_\_\_\_\_.
- The **85th percentile** is \_\_\_\_\_.
- **Median** is \_\_\_\_\_.

### Data on Pinky Length

#### Table of Pinky Length Measurements
- Measurements: 7, 7, 7, 7, 6.5, 6.5, 6.5, 6.5, 6, 6, 7, 7, 5, 6.5, 7, 6.5, 6, 6.5, 6.5, 6, 5.5, 6.5, 6, 6.5, 6.5, 6, 6, 5.5, 6, 6.5

#### Statistical Summary
- **Average (Mean):** 6.433333
- **Standard Deviation (S):** 0.486602
- **1st Quartile (Q1):** 6
- **2nd Quartile (Q2/Median):** 6.5
- **Interquartile Range (IQR):** 0.875
- **15th Percentile:** 6
- **85th Percentile:** 7
- **Normal Distribution Approximation:**
- \( X \sim \mathcal{N}(6.43, 0.487) \)

#### Probability Calculation
- **Probability of Length Greater than 6.5 cm:**
- \( P(X > 6.5) = 0.266667 \)

#### Histogram Description
The histogram visualizes the frequency distribution of pinky lengths:
- **Intervals:**
1. [5, 5.55]
2. (5.55, 6.1]
3. (6.1, 6.65]
4. (6.65, 7.2]

- **Frequency:**
- [5, 5.55]: 2
- (5.55, 6.1]: 6
- (6.1, 6.65]: 13 (most frequent)
- (6.65, 7.2]: 9

This data and its analysis allow us to understand the distribution of pinky lengths in the sample, highlighting the central tendency, variability, and the probability of certain lengths.

Definition Definition Middle value of a data set. The median divides a data set into two halves, and it also called the 50th percentile. The median is much less affected by outliers and skewed data than the mean. If the number of elements in a dataset is odd, then the middlemost element of the data arranged in ascending or descending order is the median. If the number of elements in the dataset is even, the average of the two central elements of the arranged data is the median of the set. For example, if a dataset has five items—12, 13, 21, 27, 31—the median is the third item in ascending order, or 21. If a dataset has six items—12, 13, 21, 27, 31, 33—the median is the average of the third (21) and fourth (27) items. It is calculated as follows: (21 + 27) / 2 = 24.

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