A study was conducted to assess the age at menarche among young female gymnasts. Based on commitment to the sport, gymnasts were divided into two groups: competitive gymnasts and recreational gymnasts.
| Age at menarche | |||
| Sample size (n) | Mean, y | Standard deviation, y | |
| Competitive | 16 | 13.4 | 1.3 |
| Recreational | 22 | 12.3 | 0.8 |
Assuming that age at menarche is normally distributed, which of the following is closest to the probability that a randomly chosen competitive gymnast will have onset of menarche at age ≥16?
Show Explanatory Sources
A normal (Gaussian) distribution is a symmetrical, bell-shaped distribution with a fixed percentage of observations lying within a certain distance of the mean. This distance is called the standard deviation (SD) and represents the degree of dispersion from the mean. The 68-95-99.7 rule for normal distributions states that 68% of all observations lie within 1 SD of the mean, 95% of all observations lie within 2 SDs of the mean, and 99.7% of all observations lie within 3 SDs of the mean.
For competitive gymnasts in this sample, the mean age at menarche is 13.4 years, with a SD of 1.3 years. Based on the 68-95-99.7 rule:
68% of observations lie within 1 SD: 13.4 ± 1.3 = 12.1-14.7.
95% of observations lie within 2 SDs: 13.4 ± 2.6 = 10.8-16.0.
99.7% of observations lie within 3 SDs: 13.4 ± 3.9 = 9.5-17.3.
An onset of menarche at age ≥16 years is 2 SDs from the mean; therefore, 2.5% of the observations must lie above 16 years (with 2.5% of observations below 10.8 years). The probability that a random competitive gymnast will have an onset of menarche at age ≥16 years is 0.025.
(Choices A, B, and C) The 68-95-99.7 rule states that 99.7% of observations lie within 3 SDs; therefore, 0.997 is the probability that a random competitive gymnast will have an onset of menarche between age 9.5 and 17.3 years. Similarly, 0.95 is the probability that a random competitive gymnast will have an onset of menarche between age 10.8 and 16.0 years (ie, within ± 2 SDs from the mean), and 0.68 is the probability that a random competitive gymnast will have an onset of menarche between age 12.1 and 14.7 years (ie, within ± 1 SD from the mean).
(Choices D and F) Based on the 68-95-99.7 rule, 32% (ie, 100% − 68%) of observations lie outside 1 SD from the mean, with half (ie, 32/2 = 16%) above and half (16%) below 1 SD from the mean. Therefore, 0.160 is the probability that a random competitive gymnast will have an onset of menarche at either age ≤12.1 years (ie, ≥1 SD below the mean) or at age ≥14.7 years (ie, ≥1 SD above the mean). Similarly, 0.0015 is the probability that a random competitive gymnast will have an onset of menarche at either age ≤9.5 years (ie, ≥3 SDs below the mean) or at age ≥17.3 years (ie, ≥3 SDs above the mean).
Educational objective:
In a normal (bell-shaped) distribution, 68% of all values are within 1 standard deviation (SD) of the mean; 95% are within 2 SDs of the mean; and 99.7% are within 3 SDs of the mean.