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) 1249 with a standard deviation of 32 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 that,

mean = = 1249

standard deviation = = 32

Using Empirical rule,

a) P( - < x <   + ) = 68%

= P( 1249 - 32 < x < 1249 + 32 ) = 68%

= P( 1217 < x < 1281 ) =68%

A range of years centered about the mean in which about 68% of the data will be found between 1217  and 1281 A.D.  

b) P( - 2 < x <   + 2 ) = 95%

= P( 1249 - 2 * 32 < x < 1249 + 2 * 32 ) = 95%

= P( 1249 - 64 < x < 1249 + 64 ) = 95%

=P( 1185 < x < 1313 ) = 95%

A range of years centered about the mean in which about 95% of the data will be found between 1185 and 1313 A.D.  

c) P( - 3 < x <   + 3 ) = 99.7%

= P( 1249 - 3 * 32 < x < 1249 + 3 * 32 ) = 99.7%

= P( 1249 - 96 < x < 1249 + 96 ) = 99.7%

=P( 1153 < x < 1345 ) = 99.7%

A range of years centered about the mean in which almost all the data will be found between 1153 and 1345  A.D.