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) 1256 with a standard deviation of 41 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.

Also known as 68-95-99.7 rule.
It states that almost all the data lies within three standard deviation for a normal data.
About 68% of data lies within 1 standard deviation of mean.
About 95% of data lies within two standard deviation of mean.
About 99.7% of data lies within 3 standard deviation of mean.

a) range of years centered about the mean in which about 68% of the data lies

[tex]\mu \pm 1(\sigma)\\=1256 \pm 41\\=(1215, 1297)[/tex]

68% of data will be found between 1215 years and 1297 years.

b) range of years centered about the mean in which about 95% of the data lies

[tex]\mu \pm 2(\sigma)\\=1256 \pm 2(41)\\=1256 \pm 82\\=(1174, 1338)[/tex]

95% of data will be found between 1174 years and 1338 years.

c) range of years centered about the mean in which about all of the data lies

[tex]\mu \pm 3(\sigma)\\=1256 \pm 3(41)\\=1256 \pm 123\\=(1133, 1379)[/tex]

All of data will be found between 1133 years and 1379 years.