"Unknown cultural affiliations and loss of identity at high...

"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?

"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?

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