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 estimating variance:
   • In a normal distribution, approximately 68% of the data lies within one standard - deviation ((\sigma)) of the mean, and approximately 95% of the data lies within two standard - deviations ((2\sigma)) of the mean.
   • If we assume that the graph is a normal distribution (since it has a bell - shape), and we estimate the range within which a significant portion of the data lies. A common way is to use the fact that for a normal distribution, about 95% of the data lies within the interval (\mu\pm2\sigma), where (\mu) is the mean and (\sigma) is the standard deviation.
   • Looking at the graph, if we assume the mean is around (x = 19) (the peak of the bell - curve), and we estimate that the interval containing about 95% of the data is from approximately (x = 17) to (x = 21).
   • The formula for the interval containing about 95% of the data in a normal distribution is (\mu - 2\sigma) to (\mu+2\sigma). If (\mu - 2\sigma=17) and (\mu + 2\sigma=21) and (\mu = 19) (by symmetry of the normal distribution).
   • Subtracting the first equation from the second: ((\mu + 2\sigma)-(\mu - 2\sigma)=21 - 17).
   • Simplifying the left - hand side gives (4\sigma), and the right - hand side is 4. So (4\sigma=4), then (\sigma = 1).
2. Recall the relationship between variance and standard deviation:
   • The variance ((\sigma^{2})) is related to the standard deviation ((\sigma)) by the formula (\sigma^{2}=\sigma\times\sigma).
   • Since (\sigma = 1), then the variance (\sigma^{2}=1\times1 = 1).

Answer:

A. 1