Final answer:
To find the value which separates the top 20% of the heights of 12-month-old baby boys, we need to find the z-score corresponding to that percentile and then use it to calculate the actual height value.
Step-by-step explanation:
To find the value which separates the top 20% of the heights of 12-month-old baby boys, we need to find the z-score corresponding to that percentile and then use it to calculate the actual height value. First, we find the z-score using the formula:
z = (x - μ) / σ
where x is the height value, μ is the average height, and σ is the standard deviation. We can rearrange the formula to solve for x:
x = z * σ + μ
Next, we find the z-score that corresponds to the top 20% of the distribution. This z-score is the inverse of the cumulative distribution function (CDF) for the normal distribution.
Using a z-score table or a statistical calculator, we find that the z-score corresponding to the top 20% is approximately 0.8416.
Finally, we substitute the z-score and the given values for μ and σ into the equation to solve for x:
x = 0.8416 * 2.9 + 76.4
= 78.44
Therefore, the value that separates the top 20% of the heights of 12-month-old baby boys is approximately 78.44 cm.