Final answer:
To find the probability of a female having an overhead reach greater than 214.30 cm, calculate the z-score and then use the standard normal distribution table to find the corresponding area to the right, which is approximately 11.68%.
Step-by-step explanation:
The probability that an individual adult female has an overhead reach distance greater than 214.30 cm can be calculated using the z-score formula and the standard normal distribution table. First, we find the z-score by subtracting the mean from the given value and then dividing by the standard deviation:
Z = (X - μ) / σ
Plugging in the values, we have:
Z = (214.30 - 205) / 7.8 ≈ 1.1936
Next, we consult the standard normal distribution table to find the probability corresponding to a z-score of 1.1936. However, since the tables usually give the area to the left of the z-score, we need to subtract this value from 1 to get the area to the right, which represents the probability that a randomly selected adult female has an overhead reach greater than 214.30 cm.
If the standard normal distribution table indicates that the area to the left of z=1.1936 is 0.8832, then the probability of a greater reach is:
P(Z > 1.1936) = 1 - 0.8832 = 0.1168
Therefore, the probability that an individual distance is greater than 214.30 cm is approximately 11.68%.