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 (rear) 1247 with a standard deviation of 37 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  will be found

 (b) a range of years centered about the mean in which about 95% of the data will be found

 (c) a range of years centered about the mean in which most all the data will be found between 

Answer 1

Given that,

mean = $\mu$ = 1247

standard deviation = $\sigma$ = 37

Using Empirical rule,

a) P( $\mu$ - $\sigma$ < x <  $\mu$ + $\sigma$ ) = 68%

= P( 1247 - 37 < x < 1247 + 37 ) = 68%

= P( 1210 < x < 1284 ) =68%

A range of years centered about the mean in which about 68% of the data will be found between 1210 and 1284 A.D.

b) P( $\mu$ - 2$\sigma$ < x <  $\mu$ + 2$\sigma$ ) = 95%

= P( 1247 - 2 * 37 < x < 1247 + 2 * 37 ) = 95%

= P( 1247 - 74 < x < 1247 + 74 ) = 95%

=P( 1173 < x < 1321 ) = 95%

A range of years centered about the mean in which about 95% of the data will be found between 1173 and 1321 A.D.

c) P( $\mu$ - 3$\sigma$ < x <  $\mu$ + 3$\sigma$ ) = 99.7%

= P( 1247 - 3 * 37 < x < 1247 + 3 * 37 ) = 99.7%

= P( 1247 - 111 < x < 1247 + 111 ) = 99.7%

=P( 1136 < x < 1358 ) = 99.7%

A range of years centered about the mean in which almost  all the data will be found between 1136 and 1358 A.D.