A physician routinely makes physical examinations of children. She is concerned that a three-year-old girl has a height of only 87.8 cm. Heights of three-year-old girls have a mean of 97.5 cm and a standard deviation of 6.9 cm (based on data from the National Health and Nutrition Examination Survey).
a. Use the range rule of thumb to find the maximum and minimum usual heights of three-year-old girls.
b. Based on the result, is the height of 87.8 cm unusual?
c. Should the physician be concerned?
a. mean ± 2 SD = ?
b. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
c. Your decision.
To answer these questions, we can use the range rule of thumb and z-scores to determine if the height of 87.8 cm is unusual and if the physician should be concerned.
a. The range rule of thumb states that for a roughly bell-shaped distribution, we can estimate the range of values within which most of the data falls. According to the rule, the maximum usual height can be approximated by adding 2 standard deviations to the mean, and the minimum usual height can be approximated by subtracting 2 standard deviations from the mean.
Maximum usual height = mean + (2 * standard deviation)
= 97.5 cm + (2 * 6.9 cm)
= 97.5 cm + 13.8 cm
= 111.3 cm
Minimum usual height = mean - (2 * standard deviation)
= 97.5 cm - (2 * 6.9 cm)
= 97.5 cm - 13.8 cm
= 83.7 cm
Therefore, the maximum usual height is approximately 111.3 cm and the minimum usual height is approximately 83.7 cm.
b. To determine if the height of 87.8 cm is unusual, we need to calculate the z-score of this height using the formula:
z-score = (observed value - mean) / standard deviation
z-score = (87.8 cm - 97.5 cm) / 6.9 cm
= -1.4058
The z-score of -1.4058 indicates that the height of 87.8 cm is below the mean and further below the maximum usual height. A z-score below -2 (or above +2) is generally considered unusual.
c. Based on the result, the height of 87.8 cm is below the mean and falls below the range of usual heights for three-year-old girls. Although it is not extremely unusual, the physician may have some concerns and may want to monitor the girl's growth or investigate further to rule out any underlying health issues.