Question

"Unknown cultural affiliations and loss of identity at high elevations." These are words 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. 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.

Elevation Zone	Number of Artifacts	Number Unidentified
7000-7500 ft	115	74
5000-5500 ft	140	30

Let p₁ be the population proportion of unidentified archaeological artifacts at the elevation zone 7000-7500 feet in the given archaeological area. Let p₂ be the population proportion of unidentified archaeological artifacts at the elevation zone 5000-5500 feet in the given archaeological area.

(a) Find a 90% confidence interval for p₁ – p₂. (Use 3 decimal places.)

lower limit
upper limit

On the Navajo Reservation, a random sample of 210 permanent dwellings in the Fort Defiance region showed that 52 were traditional Navajo hogans. In the Indian Wells region, a random sample of 137 permanent dwellings showed that 19 were traditional hogans. Let p₁ be the population proportion of all traditional hogans in the Fort Defiance region, and let p₂ be the population proportion of all traditional hogans in the Indian Wells region.

(a) Find a 95% confidence interval for p₁ – p₂. (Use 3 decimal places.)

lower limit
upper limit

Answer 1

Solution

First Question

Back-up Theory

Let

X = number of unidentified archaeological artifacts at the elevation zone 7000-7500 feet in the given archaeological area. number of unidentified archaeological artifacts at the elevation zone 5000-5500 feet in the given archaeological area.

Y = number of unidentified archaeological artifacts at the elevation zone 5000-5500 feet in the given archaeological area.

Then , X ~ B(n, p₁), Y ~ B(m, p₂), where p₁ and p₂ are the respective population proportions and n and m are corresponding sample sizes.

100(1 - α) % Confidence Interval for (p₁ - p₂) is:

{(x/n) - (y/m)} ± (Z_α/2)√[p_cap(1 - p_cap){(1/n) + (1/m)}]

where p_cap = (x + y)/(n + m) and Z_α/2 is the upper (α /2)% point of N(0, 1), n and m are the two sample sizes.

Now to work out the solution,

Given n = 115, m = 140, x = 74 and y = 30, p_cap = (74 + 30)/(115 + 140) = 0.4078

90% Confidence Interval for (p₁ - p₂) is:

{(74/115) - (30/140)} ± (1.645)√[ (0.4078 x 0.5922){(1/115) + (1/140}]

= 0.4292 ± 0.1017

= [0.3275, 0.5309]

Thus, Lower Limit = 0.328 and Upper Limit = 0.531 Answer 1

Second Question

Back-up Theory

Let

X = number of all traditional hogans in the Fort Defiance region, and

Y = number of of all traditional hogans in the Indian Wells region.

Then , X ~ B(n, p₁), Y ~ B(m, p₂), where p₁ and p₂ are the respective population proportions and n and m are corresponding sample sizes.

100(1 - α) % Confidence Interval for (p₁ - p₂) is:

{(x/n) - (y/m)} ± (Z_α/2)√[p_cap(1 - p_cap){(1/n) + (1/m)}]

where p_cap = (x + y)/(n + m) and Z_α/2 is the upper (α /2)% point of N(0, 1), n and m are the two sample sizes.

Now to work out the solution,

Given n = 210, m = 137, x = 52 and y = 19, p_cap = (52 + 19)/(210 + 137) = 0.2046

95% Confidence Interval for (p₁ - p₂) is:

{(52/210) - (19/137)} ± (1.96)√[ (0.2046 x 0.7954){(1/210) + (1/137}]

= 0.1083 ± 0.0868

= [0.0221, 0.1958]

Thus, Lower Limit = 0.022 and Upper Limit = 0.196 Answer 2

DONE