Question

youre analyzing the number of hours students sleep each night to study sleep habits among your classmates. the recorded hours are as follows: \\(\\{5, 6, 7, 8, 8, 8, 9, 9, 10, 10\\}\\).

should you use a normal curve to represent this data? why or why not?

no, the data is skewed and contains multiple outliers, so it does not fit a normal curve.
yes, the data shows no significant gaps between values, and the distribution appears evenly spread around the mean, indicating that a normal curve is appropriate.
yes, the data is symmetrically distributed, and the mean, median, and mode are equal, which aligns with the characteristics of a normal curve.
no, the data set is too small and lacks enough variation to use a normal curve accurately.

Explanation:

Analyze the given dataset

$$ S = \{5, 6, 7, 8, 8, 8, 9, 9, 10, 10\} $$

$$ n = 10 $$

Calculate central tendency measures

$$ \text{Mean} = \frac{5+6+7+8+8+8+9+9+10+10}{10} = \frac{80}{10} = 8 $$

$$ \text{Median} = \frac{8+8}{2} = 8 $$

$$ \text{Mode} = 8 $$

Evaluate distribution symmetry

$$ \text{Mean} = \text{Median} = \text{Mode} = 8 $$

The data is symmetrically distributed around the center value of 8, which aligns with the characteristics of a normal curve.

Answer:

• (A) No, the data is skewed and contains multiple outliers, so it does not fit a normal curve.
• (B) Yes, the data shows no significant gaps between values, and the distribution appears evenly spread around the mean, indicating that a normal curve is appropriate.
• (C) Yes, the data is symmetrically distributed, and the mean, median, and mode are equal, which aligns with the characteristics of a normal curve. (Correct answer)
• (D) No, the data set is too small and lacks enough variation to use a normal curve accurately.