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Tree-ring dates were used extensively in archaeological studies at Burnt Mesa Pueblo (Bandelier Archaeological Excavation Project: Summer 1989 Excavations at Burnt Mesa Pueblo, edited by Kohler, Washington State University Department of Anthropology). At one site on the mesa, tree-ring dates (for many samples) gave a mean date of \(\mu_{1}=\) year 1272 with standard deviation \(\sigma_{1}=35\) years. At a second, removed site, the tree-ring dates gave a mean of \(\mu_{2}=\) year 1122 with standard deviation \(\sigma_{2}=40\) years. Assume that both sites had dates that were approximately normally distributed. In the first area, an object was found and dated as \(x_{1}=\) year \(1250 .\) In the second area, another object was found and dated as \(x_{2}=\) year 1234 . (a) Convert both \(x_{1}\) and \(x_{2}\) to \(z\) values, and locate both of these values under the standard normal curve of Figure \(6-15\). (b) Which of these two items is the more unusual as an archaeological find in its location?

Step by step solution

01

Understand the Problem

Our goal is to compute the z-scores for two archaeological dates, one from each of two different sites, and determine which date is more unusual in its respective context.

02

Calculate z-score for Site 1

The formula to convert a raw score to a z-score is \( z = \frac{x - \mu}{\sigma} \). For the first site, we have \( x_1 = 1250 \), \( \mu_1 = 1272 \), and \( \sigma_1 = 35 \). Calculating, we get \[ z_1 = \frac{1250 - 1272}{35} = \frac{-22}{35} \approx -0.63 \].

03

Calculate z-score for Site 2

For the second site, we use the same formula with values \( x_2 = 1234 \), \( \mu_2 = 1122 \), and \( \sigma_2 = 40 \). Calculating, we find \[ z_2 = \frac{1234 - 1122}{40} = \frac{112}{40} = 2.80 \].

04

Interpret the z-scores

A z-score tells us how many standard deviations an observation is from the mean. The z-score of \( z_1 = -0.63 \) means the date at the first site is 0.63 standard deviations below the mean of site 1, while \( z_2 = 2.80 \) indicates the date at the second site is 2.80 standard deviations above the mean of site 2.

05

Determine Which Item is More Unusual

The item at Site 2 has a z-score of 2.80, which means it is further from the mean compared to the item at Site 1, which has a z-score of \(-0.63\). Since larger absolute z-scores indicate rarer events, the item at Site 2 is more unusual.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Archaeological Data

Archaeological data is crucial for understanding past human activities. At Burnt Mesa Pueblo, tree-ring dating helps provide chronological frameworks. Tree rings grow annually and vary based on climate, offering specific historical timelines when trees were alive. By analyzing tree rings, archaeologists can date wood samples with high precision. This method is invaluable, providing a temporal context for artifacts and structures found in excavation sites. In the scenario given,

• Tree-ring dates at two sites offered insights into time periods when the sites were active.
• They recorded mean dating years of 1272 and 1122, respectively.

Understanding archaeological data offers a glimpse into the past, and such precise dating methods aid in piecing together historical narratives.

Normal Distribution

The concept of normal distribution is foundational in statistics, describing how data points are spread around a mean. When data is normally distributed,

• it forms a symmetric, bell-shaped curve.
• Most data points fall close to the mean, with fewer data points appearing as you move further from it.

In archaeological studies like the one at Burnt Mesa Pueblo, assuming data follows a normal distribution can simplify analysis. Tree-ring dates were assumed to approximate a normal distribution. This assumption allows for standard statistical methods to be applied, facilitating the comparison of individual dates within the dataset.

Z-scores

Z-scores are essential in statistics for assessing how unusual a data point is within a dataset. They indicate how many standard deviations a particular value is from the mean. Given a mean date and standard deviation,

• a z-score transforms a raw data point into a standardized value.
• In this archaeological problem, z-scores were used to identify how typical or atypical certain radiocarbon dates were for their respective sites.

The z-score formula is given by \[ z = \frac{x - \mu}{\sigma} \]where \( x \) is the raw data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. A high positive or negative z-score suggests that the data point is quite atypical, indicating a potential outlier or unique archaeological find.

Standard Deviation

Standard deviation is a measure of the amount of variation in a set of data values. In simple terms, it tells us how much the data points deviate from the average (mean) value.

• If the standard deviation is small, it means the data points are clustered closely around the mean.
• If the standard deviation is large, the data points are more spread out.

In our exercise, the tree-ring dates had standard deviations of 35 and 40 years for the two sites, respectively. These values give us an idea of how much the tree-ring dates varied at each site over time. Calculating a standard deviation involves taking the square root of the variance, which is the average of the squared differences from the mean. Understanding standard deviation is key to interpreting the variability and reliability of data in archaeological research.