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Archaeology: Cultural Affiliation "Unknown cultural affiliations and loss of identity at high elevations." These words are used to propose the hypothesis that archaeological sites tend to lose their identity as altitude extremes are reached. This idea is based on the notion that prehistoric people tended \(n o t\) to take trade wares to temporary settings and/or isolated areas (Source: Prehistoric New Mexico: Background for Survey, by D. E. Stuart and R. P. Gauthier, University of New Mexico Press). As elevation zones of prehistoric people (in what is now the state of New Mexico) increased, there seemed to be a loss of artifact identification. Consider the following information. $$ \begin{array}{lcc} \hline \text { Elevation Zone } & \text { Number of Artifacts } & \text { Number Unidentified } \\ \hline 7000-7500 \mathrm{ft} & 112 & 69 \\ 5000-5500 \mathrm{ft} & 140 & 26 \\ \hline \end{array} $$ Let \(p_{1}\) be the population proportion of unidentified archaeological artifacts at the elevation zone \(7000-7500\) feet in the given archaeological area. Let \(p_{2}\) be the population proportion of unidentified archaeological artifacts at the elevation zone \(5000-5500\) feet in the given archaeological area. (a) Check Requirements Can a normal distribution be used to approximate the \(\hat{p}_{1}-\hat{p}_{2}\) distribution? Explain. (b) Find a \(99 \%\) confidence interval for \(p_{1}-p_{2}\). (c) Interpretation Explain the meaning of the confidence interval in the context of this problem. Does the confidence interval contain all positive numbers? all negative numbers? both positive and negative numbers? What does this tell you (at the \(99 \%\) confidence level) about the comparison of the population proportion of unidentified artifacts at high elevations \((7000-7500\) feet \()\) with the population proportion of unidentified artifacts at lower elevations (5000-5500 feet)? How does this relate to the stated hypothesis?

Step by step solution

01

Calculate Sample Proportions

First, calculate the sample proportions of unidentified artifacts for each elevation zone. Use the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of unidentified artifacts and \( n \) is the total number of artifacts.ewlineFor 7000-7500 ft:ewline\( \hat{p}_1 = \frac{69}{112} = 0.6161\) (rounded to four decimal places)ewlineFor 5000-5500 ft:ewline\( \hat{p}_2 = \frac{26}{140} = 0.1857\) (rounded to four decimal places).

02

Check Normal Approximation Requirements

Before proceeding with the confidence interval, check whether the sample sizes are large enough for a normal approximation. The conditions are \( n_1\hat{p}_1, n_1(1-\hat{p}_1) > 5 \) and \( n_2\hat{p}_2, n_2(1-\hat{p}_2) > 5 \).ewlineFor 7000-7500 ft: \( n_1 = 112, \hat{p}_1 = 0.6161 \) gives \( n_1 \hat{p}_1 \approx 69 \) and \( n_1(1-\hat{p}_1) \approx 43 \).ewlineFor 5000-5500 ft: \( n_2 = 140, \hat{p}_2 = 0.1857 \) gives \( n_2 \hat{p}_2 \approx 26 \) and \( n_2(1-\hat{p}_2) \approx 114 \).ewlineAll values are greater than 5, so the normal approximation is valid.

03

Calculate the Standard Error

The standard error (SE) for the difference in proportions \( \hat{p}_1 - \hat{p}_2 \) is given by:ewline\[ SE = \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} } \]ewlineCalculate each part:ewline\( SE = \sqrt{ \frac{0.6161 \times 0.3839}{112} + \frac{0.1857 \times 0.8143}{140} } = 0.0648 \).

04

Find the Z-value for 99% Confidence

For a 99% confidence interval, look up the z-value that corresponds to 99% in a standard normal distribution table. The z-value for 99% confidence is approximately 2.576.

05

Calculate the Confidence Interval

The confidence interval for \( \hat{p}_1 - \hat{p}_2 \) is given by:ewline\[ (\hat{p}_1 - \hat{p}_2) \pm z \times SE \]ewline\( = (0.6161 - 0.1857) \pm 2.576 \times 0.0648 \)ewline\( = 0.4304 \pm 0.1670 \)ewlineThe confidence interval is \( [0.2634, 0.5974] \).

06

Interpretation of the Confidence Interval

The confidence interval \([0.2634, 0.5974]\) indicates that, at a 99% confidence level, the difference in the population proportions of unidentified artifacts between the two elevation zones is between 0.2634 and 0.5974. Since this interval contains only positive numbers, it suggests that a higher proportion of artifacts are unidentified at the higher elevation zone of 7000-7500 feet compared to 5000-5500 feet. This supports the hypothesis that the loss of identity is more significant at higher elevations.

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Key Concepts

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

Hypothesis testing

In statistics, hypothesis testing is a method used to determine if there is enough evidence to support a particular belief or hypothesis about a population parameter. In this archaeological context, researchers propose a hypothesis that unidentified artifacts increase as altitude increases. To test such a hypothesis, you would start by stating a null hypothesis (often denoted as \(H_0\)) and an alternative hypothesis (\(H_a\)).

The null hypothesis usually suggests no effect or no difference; for instance, "there is no difference in the proportion of unidentified artifacts between the two elevation zones." The alternative hypothesis posits what you seek to demonstrate, such as "there is a higher proportion of unidentified artifacts at higher elevations."

Once the hypotheses are set, statistical tests are conducted using collected data to determine the likelihood that the null hypothesis is true. If the likelihood is sufficiently low, the null hypothesis is rejected in favor of the alternative hypothesis. This process helps inform researchers whether the observations (like more unidentified artifacts at higher altitudes) are significant and not just due to random chance.

Confidence interval

A confidence interval provides a range of values that is likely to contain the true population parameter, with a certain level of confidence. It is derived from sample data and helps researchers make inferences about a population without studying every individual within it.

To calculate a confidence interval for a difference in proportions (like unidentified artifacts at different elevations), you first determine the standard error, which measures the variability of the sample statistic. Then, using the desired confidence level (such as 99% in this case), find the corresponding z-value from statistical tables, which reflects how many standard deviations from the mean are needed to capture the central percentage of the data.

The confidence interval is computed by taking the sample difference ofthe proportions and adding and subtracting the margin of error (z-value times standard error). For example, a 99% confidence interval of \([0.2634, 0.5974]\) for the difference in unidentified artifact proportions tells us that we are 99% confident the true difference is between these values. Importantly, if a confidence interval does not contain zero, it indicates a significant difference.

Population proportion

Population proportion refers to the ratio of members of a group that possess a certain characteristic relative to the total population. In the context of archaeology, it involves the proportion of unidentified artifacts at specific elevation zones. These proportions are crucial for comparing different groups within a study.

The sample proportion, denoted as \(\hat{p}\), is calculated as the number of items with the characteristic of interest (e.g., unidentified artifacts) divided by the total number of items sampled (e.g., all artifacts examined). This value provides an estimate of the population proportion, \(p\), for that specific characteristic.

Knowing the population proportions at varying elevations allows researchers to determine if differences in artifact identification are statistically significant. If one elevation zone shows a much higher proportion of unidentified artifacts, as seen in the interval \(p_1 - p_2\) for two zones, this could indicate environmental or historical factors affecting artifact preservation or identification. Tracking these proportions assists researchers in understanding how and why cultural affiliations might have shifted across different geographical or climactic landscapes.