A 14 yr old boy is 66 inches tall. His 6 yr old brother is 47 inches tall. Assume the heights of boys are normally distributed.
Age: 6 Mean Height: 45.8 inches Standard Deviation: 1.4
Age: 14 Mean Height: 64.3 inches Standard Deviation: 2.8
a) Which boy is taller for his age? Explain.
b) If there are 11 other boys in the 6 yr old’s class at school, how many of them would you expect to be taller than he is?
The correct answer and explanation is :
iturn0image0turn0image2turn0image6turn0image11To assess the heights of the 14-year-old and 6-year-old boys relative to their age groups, we can calculate their z-scores. A z-score indicates how many standard deviations a particular value is from the mean. The formula for the z-score is:
[ z = \frac{X – \mu}{\sigma} ]
where:
- ( X ) is the individual’s height,
- ( \mu ) is the mean height for that age group,
- ( \sigma ) is the standard deviation for that age group.
a) Which boy is taller for his age?
For the 14-year-old boy:
- Height (( X )) = 66 inches
- Mean height (( \mu )) = 64.3 inches
- Standard deviation (( \sigma )) = 2.8 inches
Calculating the z-score:
[ z = \frac{66 – 64.3}{2.8} = \frac{1.7}{2.8} \approx 0.607 ]
This z-score of approximately 0.607 means the 14-year-old boy’s height is about 0.607 standard deviations above the mean for his age group.
For the 6-year-old boy:
- Height (( X )) = 47 inches
- Mean height (( \mu )) = 45.8 inches
- Standard deviation (( \sigma )) = 1.4 inches
Calculating the z-score:
[ z = \frac{47 – 45.8}{1.4} = \frac{1.2}{1.4} \approx 0.857 ]
This z-score of approximately 0.857 indicates the 6-year-old boy’s height is about 0.857 standard deviations above the mean for his age group.
Conclusion: The 6-year-old boy has a higher z-score (0.857) compared to the 14-year-old boy (0.607), indicating that, relative to their respective age groups, the 6-year-old is taller for his age.
b) If there are 11 other boys in the 6-year-old’s class at school, how many of them would you expect to be taller than he is?
A z-score of 0.857 corresponds to a percentile rank of approximately 80.5%. This means the 6-year-old boy is taller than about 80.5% of boys his age. Consequently, approximately 19.5% of boys his age are taller than he is.
In a class with 11 other boys:
[ \text{Number of boys taller} = 11 \times 0.195 \approx 2.145 ]
Rounding to the nearest whole number, we would expect about 2 of the 11 other boys to be taller than the 6-year-old boy.
Explanation:
The z-score is a statistical measurement that describes a value’s position relative to the mean of a group of values, measured in terms of standard deviations. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean. By converting individual heights into z-scores, we can compare how each boy’s height relates to his age group’s average height, even though the age groups have different means and standard deviations.
Percentiles indicate the relative standing of a value within a dataset. For example, being in the 80.5th percentile means the individual is taller than 80.5% of the population. In this context, knowing the percentile helps estimate how many peers are taller or shorter than the individual.
By applying these statistical tools, we can objectively assess and compare the boys’ heights relative to their age groups and estimate how many peers might be taller or shorter.