Question

at blunt 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 (years) 1244 with a standard deviation of 46 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 foundbetween and a.d.(b) a range of years centered about the mean in which about 95% of the data (tree - ring dates) will be foundbetween and a.d.(c) a range of years centered about the mean in which almost all the data (tree - ring dates) will be foundbetween and a.d.

Explanation:

Step1: Recall the empirical rule

The empirical rule for a normal distribution (mound - shaped and symmetric) states:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\))
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\sigma\)) of the mean (\(\mu\))
• Approximately \(99.7\%\) (almost all) of the data lies within \(3\) standard deviations (\(\sigma\)) of the mean (\(\mu\))

Given \(\mu = 1244\) and \(\sigma=46\)

Step2: Calculate for \(68\%\)

For \(68\%\) of the data:
The lower bound is \(\mu-\sigma\) and the upper bound is \(\mu + \sigma\)

$$
LATEXBLOCK0
$$

Step3: Calculate for \(95\%\)

For \(95\%\) of the data:
The lower bound is \(\mu - 2\sigma\) and the upper bound is \(\mu+2\sigma\)

$$
LATEXBLOCK1
$$

Step4: Calculate for \(99.7\%\) (almost all)

For \(99.7\%\) of the data:
The lower bound is \(\mu-3\sigma\) and the upper bound is \(\mu + 3\sigma\)

$$
LATEXBLOCK2
$$

Answer:

(a) Between \(1198\) and \(1290\) A.D.
(b) Between \(1152\) and \(1336\) A.D.
(c) Between \(1106\) and \(1382\) A.D.