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

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.

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

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.