Solutions:

Mean of the given data set: $10, 12, 9, 8, 11, 9, 6, 12, 10, 7, 8, 6, 12, 9, 8$

1. Calculate the sum of all values: $10 + 12 + 9 + 8 + 11 + 9 + 6 + 12 + 10 + 7 + 8 + 6 + 12 + 9 + 8 = 137$

2. Count the number of values: $15$

3. Calculate the mean: $\text{Mean} = \frac{137}{15} \approx 9.13$

Shape of the given data set: The shape of a data set can be described as left-skewed, symmetrical, or right-skewed based on the distribution of its values.

By inspecting the given values: $10, 12, 9, 8, 11, 9, 6, 12, 10, 7, 8, 6, 12, 9, 8$

We notice:

• Median: 9
• Mean: 9.13

Since the mean is slightly greater than the median, the data set is approximately symmetrical.

Range of the given data set: The range is the difference between the maximum and minimum values.

1. Identify the maximum and minimum values: $\text{Max} = 12, \text{Min} = 6$

2. Calculate the range: $\text{Range} = 12 - 6 = 6$

Variance of the given data set:

1. Calculate the mean ($\mu$): $\mu \approx 9.13$

2. Calculate each value's deviation from the mean, square it, and find the average of these squared deviations:

$\frac{(10 - 9.13)^2 + (12 - 9.13)^2 + (9 - 9.13)^2 + \cdots + (8 - 9.13)^2}{15} \approx 4.39$

Does NOT belong to the group: Interquartile range, 50th Percentile, Second Quartile, Median

All these terms refer to the central tendency and spread of the data set and are equivalent measures.

True statement:

1. $z$-score helps identify outliers in a set of data.
2. If the mean < median, the data are negatively skewed (right-skewed).
3. Another term for the independent variable is the predictor variable.

NOT True statement:

• Interval has no true zero value.
• Parametric refers to numerical characteristics that describe a population.
• Descriptive statistics summarize and present data in graphical or textual form.

IS True statement:

• Statistic refers to the numerical characteristic that describes a sample.

Questions:

1. How is the standard deviation related to variance?
2. What is the difference between the mean and the median?
3. How can you identify if a data set is skewed?
4. What are some real-world examples of right-skewed distributions?
5. Why is the median often used instead of the mean in skewed distributions?
6. What are the key differences between descriptive and inferential statistics?
7. How do you interpret a $z$-score?
8. What is the purpose of calculating the range of a data set?

Tip:

Always use appropriate statistical methods to analyze and interpret data to make informed decisions and predictions.