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) 1239 with a standard deviation of 43 years. The distribution of dates was more or less mound-shaped and symmetrical 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 that ,

mean = = 1239

standard deviation = = 43

Using Empirical rule,

a) P( - < x <   + ) = 68%

= P( 1239 - 43 < x < 1239 + 43 ) = 68%

= P( 1196 < x < 1282 ) =68%

a range of years centered about the mean in which about 68% of the data (tree-ring dates) will be found
between 1196 and 1282 A.D

b) P( - 2 < x <   + 2 ) = 95%

= P( 1239 - 2 * 43 < x < 1239 + 2 * 43 ) = 95%

= P( 1239 - 86 < x < 1239 + 86) = 95%

=P( 1153 < x < 1325 ) = 95%

a range of years centered about the mean in which about 95% of the data (tree-ring dates) will be found
between 1153 and 1325 A.D

c) P( - 3 < x <   + 3 ) = 99.7%

= P( 1239 - 3 * 43 < x < 1239 + 3 * 43 ) = 99.7%

= P( 1239 - 129 < x < 1239 + 129 ) = 99.7%

=P( 1110 < x < 1368 ) = 99.7%

a range of years centered about the mean in which about 99.7% of the data (tree-ring dates) will be found
between 1110 and 1368 A.D