Question

check all of the situations that you would expect to be approximately normally distributed

the sat scores of high school seniors
the salaries for members of a professional football team
the shoe sizes of 17-year-old males
the age at which a child loses his or her first tooth
the probability of rolling a given number between 1 and 6 on a number cube

Explanation:

Analyze the characteristics of a normal distribution

A normal distribution is a symmetric, bell-shaped curve where most observations cluster around the central peak (the mean), and the probabilities taper off equally in both directions. It typically models natural biological measurements, standardized test scores, and physical characteristics of a homogeneous population where extreme values are rare and equally likely on both sides of the average.

Evaluate each given situation

• the SAT scores of high school seniors: Standardized test scores like the SAT are designed by test-makers to be approximately normally distributed, with most students scoring near the middle and fewer students scoring extremely high or extremely low. This is a classic example of a normal distribution.
• the salaries for members of a professional football team: Salaries in professional sports are highly skewed to the right. A few superstar players earn exceptionally high multi-million dollar salaries, while the majority of the roster earns closer to the league minimum. This creates a highly asymmetric, right-skewed distribution.
• the shoe sizes of 17-year-old males: Biological measurements (like height, weight, and shoe size) for a specific, homogeneous demographic group (17-year-old males) naturally cluster around an average value and taper off symmetrically. This is approximately normally distributed.
• the age at which a child loses his or her first tooth: This biological milestone occurs for almost all children within a relatively tight, symmetric age range (typically between 5 and 7 years old), with most children losing it around age 6. This biological developmental variable is approximately normally distributed.
• the probability of rolling a given number between 1 and 6 on a number cube: Each outcome on a fair six-sided die has an equal probability of \(\frac{1}{6}\). This represents a discrete uniform distribution, not a bell-shaped normal distribution.

Identify the correct selections

Based on the analysis, the situations that are expected to be approximately normally distributed are:

1. the SAT scores of high school seniors
2. the shoe sizes of 17-year-old males
3. the age at which a child loses his or her first tooth

Answer:

• the SAT scores of high school seniors (Correct answer)
• the salaries for members of a professional football team
• the shoe sizes of 17-year-old males (Correct answer)
• the age at which a child loses his or her first tooth (Correct answer)
• the probability of rolling a given number between 1 and 6 on a number cube