the graph shows the distribution of hand lengths, in centimeters, of the members of an orchestra. hand…

1. Recall the properties of a normal - distribution graph for variance determination:
   • In a normal distribution, if the graph is symmetric about the mean and we assume that the points at approximately one - standard deviation away from the mean are clearly marked. In a normal distribution, the empirical rule states that about 68% of the data lies within one standard deviation ((\sigma)) of the mean.
   • Looking at the graph, if we assume the mean is at (x = 19) (the peak of the normal - distribution curve), and the points that seem to mark the boundaries of the middle 68% of the data are around (x = 17) and (x = 21).
   • The formula for the relationship between the standard deviation ((\sigma)) and the data points in a normal distribution for the 68 - 95 - 99.7 rule is that the interval ((\mu-\sigma,\mu + \sigma)) contains about 68% of the data. If (\mu) is the mean and (\mu = 19), and (\mu-\sigma=17) and (\mu+\sigma = 21), then we can solve for (\sigma).
   • From (\mu-\sigma=17) and (\mu = 19), we substitute (\mu) into the equation: (19-\sigma=17), so (\sigma = 2).
2. Recall the formula for variance ((\sigma^{2})):
   • The variance ((\sigma^{2})) is related to the standard deviation ((\sigma)) by the formula (\sigma^{2}=\sigma\times\sigma).
   • Since (\sigma = 2), then (\sigma^{2}=4).

Answer:

4