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"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 not 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. 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) 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) 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

Check Normality Conditions for Approximating Distribution

To determine whether a normal distribution can be used to approximate \( \hat{p}_{1} - \hat{p}_{2} \), we need to verify the conditions for normal approximation of the difference between two proportions. These conditions are \( n_{1} \hat{p}_{1} > 5 \), \( n_{1} (1 - \hat{p}_{1}) > 5 \), \( n_{2} \hat{p}_{2} > 5 \), and \( n_{2} (1 - \hat{p}_{2}) > 5 \), where \( n_{1} \) and \( n_{2} \) are the sample sizes at respective elevation zones. You must check these calculations to confirm the conditions are met. Without specific numbers, we assume both conditions are fulfilled based on typical archaeological sample sizes.

02

Calculate Confidence Interval for Difference of Proportions

Given sample size and proportions, the confidence interval for \( p_{1} - p_{2} \) can be calculated using the formula: \[(\hat{p}_{1} - \hat{p}_{2}) \pm Z \times \sqrt{\frac{\hat{p}_{1}(1 - \hat{p}_{1})}{n_{1}} + \frac{\hat{p}_{2}(1 - \hat{p}_{2})}{n_{2}}}\]Where \( Z \) is the critical value for a 99% confidence level (approximately 2.576). Substitute the sample statistics into the formula to calculate the interval.

03

Interpret Confidence Interval

The confidence interval represents a range for the difference in the proportion of unidentified artifacts between the two elevation zones. If the interval includes zero, we conclude that there's no significant difference between the proportions. If entirely positive, \( p_{1} > p_{2} \); if entirely negative, \( p_{1} < p_{2} \). This result indicates whether higher elevations (7000-7500 feet) have a higher or lower proportion of unidentified artifacts compared to lower elevations (5000-5500 feet), reflecting the relationship to the hypothesis regarding artifact identity and elevation.

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Key Concepts

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

Confidence Intervals

Confidence intervals are a powerful statistical tool used to estimate the range in which a population parameter, such as the difference between two proportions, lies. In the archaeological example provided, a confidence interval allows us to estimate the range for the difference in the proportions of unidentified artifacts between two elevation zones (7000-7500 feet and 5000-5500 feet).

To calculate the confidence interval, we use sample statistics – specifically, the sample proportions at both elevation zones – and apply a formula that incorporates the critical value for the desired confidence level. For a 99% confidence level, the critical value (Z) is approximately 2.576. This level of confidence suggests that there is a 99% probability that the calculated interval contains the true difference of proportions.

When interpreting the confidence interval:

• If the interval includes zero, it suggests no significant difference between the two proportions.
• If the interval contains only positive numbers, it indicates that the proportion of unidentified artifacts is likely higher at the 7000-7500 feet elevation.
• Conversely, if it contains only negative numbers, the proportion is likely higher at the 5000-5500 feet elevation.

Through examining whether only positive or negative numbers are present, researchers can infer about the relationship between elevation and artifact identification, in line with the hypothesis concerning cultural affiliations at extreme elevations.

Normal Distribution

The normal distribution is a bell-shaped curve that is symmetrical about the mean and is fundamental in statistics for approximating distributions of sample data under certain conditions. When discussing the difference of proportions, such as in our archaeological example, we use the normal distribution to estimate how the difference behaves under repeated sampling.

To justify the use of a normal distribution for the difference between two sample proportions, specific conditions must be met. These conditions involve ensuring that both the expected number of successes and failures in each sample are greater than five, as expressed in the criteria:

• \( n_{1} \hat{p}_{1} > 5 \)
• \( n_{1} (1 - \hat{p}_{1}) > 5 \)
• \( n_{2} \hat{p}_{2} > 5 \)
• \( n_{2} (1 - \hat{p}_{2}) > 5 \)

Assuming these conditions are met, we can use the normal distribution to approximate the behavior of the difference between sample proportions \( \hat{p}_{1} - \hat{p}_{2} \).

This approximation allows us to calculate confidence intervals and make inferences about the populations from which the samples were drawn, assessing the validity of hypotheses such as those concerning loss of cultural identity with elevation.

Difference of Proportions

In statistics, the difference of proportions compares two independent proportions to determine if there is a statistically significant difference between them. This is particularly important when testing hypotheses, such as whether the elevation affects the proportion of unidentified artifacts in archaeological studies.

The formula used for the confidence interval of the difference of proportions incorporates both the sample proportions and the sample sizes. Mathematically, it is represented as:\[(\hat{p}_{1} - \hat{p}_{2}) \pm Z \times \sqrt{\frac{\hat{p}_{1}(1 - \hat{p}_{1})}{n_{1}} + \frac{\hat{p}_{2}(1 - \hat{p}_{2})}{n_{2}}}\]
The resulting interval helps us understand which population proportion is larger and by how much, within a certain level of confidence.A well-calculated difference – supported by statistically significant results – provides insight into how different conditions (such as elevation) might impact the unidentified artifacts. This not only helps substantiate the hypothesis but also enhances our understanding of cultural identity dynamics across various environmental factors.