Question

At Burnt Mesa Pueblo, archaeological studies have used the method of tree-ring dating in an effort to determine when prehistoric people lived in the pueblo. Wood from several excavations gave a mean of (year) 1257 with a standard deviation of 38 years. The distribution of dates was more or less mound-shaped and symmetric about the mean. Use the empirical rule to estimate the following.

(a) a range of years centered about the mean in which about 68% of the data (tree-ring dates) will be found
between  and  A.D.

(b) a range of years centered about the mean in which about 95% of the data (tree-ring dates) will be found
between  and  A.D.

(c) a range of years centered about the mean in which almost all the data (tree-ring dates) will be found
between  and  A.D.

Answer 1

Solution:

Given, the Normal distribution with,

   = 1257

= 38

According to empirical rule:

68% observations lie between - and   + .i.e within one standard deviation of the mean

95% observations lie between - 2 and   + 2 .i.e within two standard deviation of the mean

99.7 (almost all) observations lie between - 3 and   + 3 .i.e within three standard deviation of the mean

a) - and   +   

1257 - 38 and 1257 + 38

between 1222 and 1295

b) - 2 and   + 2  

1257 - 2*38 and 1257 + 2*38

between 1181 and 1333

c) - 3 and   + 3  

1257 - 3*38 and 1257 + 3*38

between 1143 and 1371