EX 3.17 Apgar scale

• Scenario:
  • Each newborn child is rated on the Apgar scale immediately after birth.
  • Possible ratings range from $0$ to $10$.
• The Apgar scale is a quick assessment tool used to evaluate the health of newborns.
  • It consists of five criteria, each scored from 0 to 2:
  1. Appearance: Skin color
     • 0: Blue or pale all over
     • 1: Pink body with blue extremities
     • 2: Completely pink
  2. Pulse: Heart rate
     • 0: No pulse
     • 1: Less than 100 beats per minute
     • 2: Over 100 beats per minute
  3. Grimace: Reflex response
     • 0: No response
     • 1: Grimace or weak cry when stimulated
     • 2: Crying and active response to stimulation
  4. Activity: Muscle tone
     • 0: Limp
     • 1: Some flexion of arms and legs
     • 2: Active movement
  5. Respiration: Breathing effort
     • 0: Not breathing
     • 1: Weak, irregular breathing
     • 2: Good, strong cry
  • Total Score:
    • The scores from each category are summed, with a maximum possible score of 10.
    • Higher scores indicate better health.
• Random variable definition:
  • Let $X$ denote the Apgar score of a randomly selected child born at a certain hospital during the next year.
• Probability mass function (pmf) of $X$:
  • The pmf provides the probabilities associated with each possible value of $X$.
  • Possible values of $X$ are:
    • $X=0,1,2,\dots ,10$

x	0	1	2	3	4	5	6	7	8	9	10
p(x)	.002	.001	.002	.005	.02	.04	.18	.37	.25	.12	.01

Then the mean value of $X$ is

$$\begin{array}{rl}E(X)& =\mu \\ & =0(0.002)+1(0.001)+2(0.002)+\cdots \\ & +8(0.25)+9(0.12)+10(0.01)\\ & =7.15\end{array}$$

• Context of the Apgar score:
  • The Apgar score $X$ can take values from $0$ to $10$.
  • While the score $\mu$ (the mean) is a calculated value, it is not a possible score for an individual child.
• Conceptual population:
  • The pmf serves as a model for a conceptual population that includes all possible values of $X$:
  $$\{0,1,2,\dots ,10\}$$
  • This conceptual population does not refer to an existing set of newborns but is based on the distribution of scores across many hypothetical future children.
• Mean of the conceptual population:
  • The mean value of this conceptual population is given by:
  $$\mu =7.15$$
  • This value reflects the average Apgar score one would expect if many children were rated, providing insight into overall health assessments.
• Interpretation:
  • While the mean $\mu$ is not an achievable score for an individual, it helps summarize the expected outcome for a large group based on the distribution modeled by the pmf.