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) 1259 with a standard deviation of 29 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 = = 1259

standard deviation = = 29

Using Empirical rule,

a) P( - < x <   + ) = 68%

= P( 1259 - 29 < x < 1259 + 29 ) = 68%

= P( 1230 < x < 1288 ) =68%

A range of years centered about the mean in which about 68% of the data (tree-ring dates) will be found between 1230 and 1288 A.D.  

b) P( - 2 < x <   + 2 ) = 95%

= P( 1259 - 2 * 29 < x < 1259 + 2 * 29 ) = 95%

= P( 1259 - 58 < x < 1259 + 58 ) = 95%

=P( 1201 < x < 1317 ) = 95%

A range of years centered about the mean in which about 95% of the data (tree-ring dates) will be found between 1201 and 1317 A.D.  

c) P( - 3 < x <   + 3 ) = 99.7%

= P( 1259 - 3 * 29 < x < 1259 + 3 * 29 ) = 99.7%

= P( 1259 - 87 < x < 1259 + 87 ) = 99.7%

=P( 1172 < x < 1346 ) = 99.7%

A range of years centered about the mean in which almost all the data (tree-ring dates) will be found between 1172 and 1346 A.D.