https://file.aiquickdraw.com/mathbot/file/1730285847491zrw5tu6z.jpg

Let's analyze each part of this exercise based on the information provided in the image.

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### Exercise 11.4 Solutions

#### (a) **Identify the Type of Distribution**

For each histogram:
1. **Histogram (1):** 
   - This histogram has a tail on the right side, indicating that it is **positively skewed**.
   - **Reason:** In positively skewed distributions, most values cluster on the left with a few high values, resulting in a right tail.

2. **Histogram (2):**
   - This histogram has a tail on the left side, indicating that it is **negatively skewed**.
   - **Reason:** In negatively skewed distributions, most values cluster on the right with a few low values, resulting in a left tail.

#### (b) **Data Analysis**

Given data values: 1, 2, 6, 8, 8, 8, 8, 8, 10, 10.

1. **Draw a Box and Whisker Plot:**
   - Calculate the quartiles:
     - **Q1**: 6
     - **Q2 (Median)**: 8
     - **Q3**: 8
   - The box and whisker plot will have:
     - Minimum = 1, Q1 = 6, Median = 8, Q3 = 8, and Maximum = 10.

2. **Symmetry Explanation:**
   - This data set is symmetric around the median (8) as the values on both sides of the median are balanced.

3. **Calculate the Mean:**
   - Mean = \( \frac{1 + 2 + 6 + 8 + 8 + 8 + 8 + 8 + 10 + 10}{10} = 6.9 \)
   - **Observation:** The mean (6.9) is slightly different from the median due to the presence of the outliers (1 and 2) on the lower end.

#### (c) **Class Scores Analysis**

Given scores: 101, 90, 85, 97, 89, 85, 84, 88, 83, 96, 93, 81, 88, 92, 88, 96, 92, 77, 85, 91, 81, 80, 87, 88.

1. **Calculate Mean, Median, and Mode:**
   - **Mean:** Add all values and divide by the number of scores.
   - **Median:** Arrange the scores and find the middle value.
   - **Mode:** Identify the most frequently occurring score(s).

2. **Draw a Box and Whisker Plot:**
   - Calculate the quartiles and represent the data distribution.

3. **Describe the Distribution:**
   - Observe if the data is symmetrical, positively skewed, or negatively skewed.

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Would you like a deeper breakdown of any part, or have specific questions about these steps?

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### Related Questions
1. How do you calculate the quartiles for a dataset?
2. What does it mean for a dataset to be symmetrical?
3. Why might a dataset have a different mean and median?
4. How does skewness affect the interpretation of data?
5. How can outliers influence the mean and median in a dataset?

#### Tip:
When comparing mean and median, remember that in a positively skewed distribution, the mean is usually greater than the median, while in a negatively skewed distribution, the mean is typically less than the median.

Based on the image, solve the following exercises related to skewed distributions, box and whisker plots, and statistical analysis.

Solve exercises related to skewed distributions and box plots. Learn to identify positively and negatively skewed distributions, construct a box and whisker plot, and analyze mean, median, and mode. Suitable for students in Grades 10-12.

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