91Ó°ÊÓ

Navajo Culture: Traditional Hogans S. C. Jett is a professor of geography at the University of California, Davis. He and a colleague, V. E. Spencer, are experts on modern Navajo culture and geography. The following information is taken from their book Navajo Architecture: Forms, History, Distributions (University of Arizona Press). On the Navajo Reservation, a random sample of 210 permanent dwellings in the Fort Defiance region showed that 65 were traditional Navajo hogans. In the Indian Wells region, a random sample of 152 permanent dwellings showed that 18 were traditional hogans. Let \(p_{1}\) be the population proportion of all traditional hogans in the Fort Defiance region, and let \(p_{2}\) be the population proportion of all traditional hogans in the Indian Wells region. (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 Examine the confidence interval and comment on its meaning. Does it include numbers that are all positive? all negative? mixed? What if it is hypothesized that Navajo who follow the traditional culture of their people tend to occupy hogans? Comment on the confidence interval for \(p_{1}-p_{2}\) in this context.

Step by step solution

01

Identify Sample Proportions

Calculate the sample proportions \( \hat{p}_1 \) and \( \hat{p}_2 \) for each region. \( \hat{p}_1 = \frac{65}{210} \approx 0.3095 \) and \( \hat{p}_2 = \frac{18}{152} \approx 0.1184 \).

02

Check Normality Requirements

To use the normal approximation for \( \hat{p}_1 - \hat{p}_2 \), check if both \( n_1 \hat{p}_1, n_1(1 - \hat{p}_1), n_2 \hat{p}_2, \) and \( n_2(1 - \hat{p}_2) \) are greater than 5. Compute:- \( n_1 \hat{p}_1 = 210 \times 0.3095 \approx 65 \)- \( n_1(1 - \hat{p}_1) = 210 \times (1 - 0.3095) \approx 145 \)- \( n_2 \hat{p}_2 = 152 \times 0.1184 \approx 18 \)- \( n_2(1 - \hat{p}_2) = 152 \times (1 - 0.1184) \approx 134 \)All values are greater than 5, so normal approximation is valid.

03

Calculate Standard Error

Calculate the standard error (SE) of \( \hat{p}_1 - \hat{p}_2 \) using:\[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]Substitute the values:\[ SE = \sqrt{\frac{0.3095 \times 0.6905}{210} + \frac{0.1184 \times 0.8816}{152}} \approx 0.0598 \]

04

Find Z-value for 99% Confidence Interval

The Z-value for a 99% confidence interval is approximately 2.576. This value is derived from standard normal distribution tables.

05

Compute Confidence Interval

Compute the confidence interval as follows:\[ \hat{p}_1 - \hat{p}_2 \pm Z \times SE \]\[ 0.3095 - 0.1184 \pm 2.576 \times 0.0598 \]\[ 0.1911 \pm 0.1541 \]This gives the interval \( (0.0370, 0.3452) \).

06

Interpretation of Confidence Interval

The confidence interval for \( p_1 - p_2 \) is (0.0370, 0.3452). Since the interval is all positive, it suggests that the proportion of traditional hogans is likely higher in the Fort Defiance region than in the Indian Wells region. This could indicate a stronger presence of traditional cultural practices in Fort Defiance.

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Key Concepts

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

Population Proportion

The population proportion, denoted as \( p \), represents the fraction of a population that possesses a particular characteristic. In statistics, when we talk about comparing two populations, such proportions are crucial. In the exercise provided, we consider two regions, Fort Defiance and Indian Wells, to determine how many traditional Navajo hogans exist relative to the total number of dwellings in each region.
\[ p_1 = \text{Proportion of traditional hogans in Fort Defiance} \]
\[ p_2 = \text{Proportion of traditional hogans in Indian Wells} \]
To calculate the sample proportions, divide the number of traditional hogans by the total number of dwellings. For instance:

• In Fort Defiance: \( \hat{p}_1 = \frac{65}{210} \approx 0.3095 \)
• In Indian Wells: \( \hat{p}_2 = \frac{18}{152} \approx 0.1184 \)

This method helps us estimate the true population proportions, \( p_1 \) and \( p_2 \). When comparing two proportions, it's often necessary to use these sample proportions to infer insights about the populations they represent.

Confidence Interval

A confidence interval provides a range of values that likely contains the true difference between two population proportions. In simpler terms, we can say with a certain degree of confidence that the interval contains the real difference between \( p_1 \) and \( p_2 \).
For this exercise, we're aiming for a 99% confidence interval for the difference between two proportions \((p_1 - p_2)\). This means we're 99% confident that the real difference between the proportions of traditional hogans in Fort Defiance and Indian Wells lies within this interval.
Using the sample proportions and the standard error computed early on, and a Z-value derived from the normal distribution for 99% confidence which is approximately 2.576, the confidence interval is calculated as:
\[ \hat{p}_1 - \hat{p}_2 \pm 2.576 \times SE \]
This resulted in the confidence interval: \((0.0370, 0.3452)\). This all-positive interval indicates that proportion values likely show Fort Defiance favoring more traditional hogans than Indian Wells.

Normal Approximation

The normal approximation helps in cases where a normal distribution curve is a good estimate for binomial distributions, especially useful when calculating confidence intervals. For using a normal approximation, certain criteria of sample sizes and proportions must be met.
In our example, to check if the distribution \( \hat{p}_1 - \hat{p}_2 \) can be considered normal, we must ensure:

• \( n_1 \hat{p}_1 > 5 \)
• \( n_1(1 - \hat{p}_1) > 5 \)
• \( n_2 \hat{p}_2 > 5 \)
• \( n_2(1 - \hat{p}_2) > 5 \)

When substituting the given values, all conditions are satisfied:

• Fort Defiance: \(65\) and \(145\)
• Indian Wells: \(18\) and \(134\)

All these figures are greater than 5, permitting the normal approximation to be used, thus allowing the computation of the confidence interval using this approximation.

Sample Size

The sample size plays a critical role in statistical analyses. It affects the reliability and accuracy of estimates, like population proportions and confidence intervals. In the exercise, we have two sample sizes:
\[ n_1 = 210 \text{ (Fort Defiance)} \]
\[ n_2 = 152 \text{ (Indian Wells)} \]
These sizes are important for ensuring the assumptions for normal approximation are met and influence the precision of our confidence interval.
Larger sample sizes generally result in more precise and narrower confidence intervals. This means our estimation of the population proportion difference will be more accurate. Conversely, smaller samples tend to enlarge the intervals, reducing confidence in the outcome. Thus in surveys or studies, choosing an adequate sample size is essential to gain reliable and meaningful statistics.

Standard Error

The standard error (SE) is a statistic that describes how much a sample mean or proportion might differ from the true population mean or proportion. It is crucial when calculating a confidence interval.
In this exercise, the SE of the difference between sample proportions \( \hat{p}_1 - \hat{p}_2 \) is needed. It is calculated with:
\[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]
By substituting the values directly:
\[ SE = \sqrt{\frac{0.3095 \times 0.6905}{210} + \frac{0.1184 \times 0.8816}{152}} \approx 0.0598 \]
This figure quantifies variability and aids in setting the width of the confidence interval for the difference between population proportions. A smaller SE suggests less variability and thus a more accurate confidence interval, whereas a larger SE implies higher variability, leading to a broader interval.