Question

the following data set represents the number of hours students from different classes spent on social media per week. the data is not appropriate for using a normal curve.

data set \\(\\{1, 2, 4, 4, 4, 5, 5, 6, 6, 7, 8, 8, 12, 15, 25\\}\\).

based on this information, evaluate the statements below and determine whether the justification is justified (i.e., it correctly supports the use of a normal curve) or not justified (i.e., the information provided is misleading or incorrect).

• the mean, median, and mode are all very close in value, indicating no skewness. (justified / not justified)
• the data shows no outliers, and most values are centered around the mean, making it symmetrical. (justified / not justified)
• there are extreme values that significantly affect the mean, leading to a skewed distribution. (justified / not justified)
• the data is heavily skewed to the right, with a long tail of values stretching toward higher numbers. (justified / not justified)

Explanation:

Calculate summary statistics

Using the Sample Mean knowledge point, we sum the 15 data values and divide by 15:

$$ \text{Mean} = \frac{1+2+4+4+4+5+5+6+6+7+8+8+12+15+25}{15} = \frac{112}{15} \approx 7.47 $$

The median is the 8th value:

$$ \text{Median} = 6 $$

The mode is the most frequent value:

$$ \text{Mode} = 4 $$

Analyze distribution skewness

Using the Skewness knowledge point, we compare the mean, median, and mode:

$$ \text{Mean } (7.47) > \text{Median } (6) > \text{Mode } (4) $$

This positive difference indicates right skewness. The values 12, 15, and 25 pull the mean upward, creating a long right tail.

Evaluate statements 1 and 2

Using the Normal Distribution Suitability knowledge point, we evaluate the first two claims:

• Statement 1: "The mean, median, and mode are all very close in value..." is false because they differ significantly (\(7.47

eq 6
eq 4\)). Thus, this is Not Justified.

• Statement 2: "The data shows no outliers, and most values are centered..." is false. The value 25 is an outlier, and the data is skewed, not symmetrical. Thus, this is Not Justified.

Evaluate statements 3 and 4

Using the Normal Distribution Suitability knowledge point, we evaluate the remaining claims:

• Statement 3: "There are extreme values that significantly affect the mean, leading to a skewed distribution." This is true and correctly explains why a normal curve is inappropriate. Thus, this is Justified.
• Statement 4: "The data is heavily skewed to the right, with a long tail..." This is true and correctly supports why a normal curve is inappropriate. Thus, this is Justified.

Answer:

Question 1

• Statement: "The mean, median, and mode are all very close in value, indicating no skewness."
• Evaluation: Not Justified

Question 2

• Statement: "The data shows no outliers, and most values are centered around the mean, making it symmetrical."
• Evaluation: Not Justified

Question 3

• Statement: "There are extreme values that significantly affect the mean, leading to a skewed distribution."
• Evaluation: Justified

Question 4

• Statement: "The data is heavily skewed to the right, with a long tail of values stretching toward higher numbers."
• Evaluation: Justified