91Ó°ÊÓ

Navajo Lifestyle: Traditional Hogans A random sample of 5222 permanent dwellings on the entire Navajo Indian Reservation showed that 1619 were traditional Navajo hogans (Navajo Architecture: Forms, History, Distributions by Jett and Spencer, University of Arizona Press). (a) Let \(p\) be the proportion of all permanent dwellings on the entire Navajo Reservation that are traditional hogans. Find a point estimate for \(p\). (b) Find a \(99 \%\) confidence interval for \(p .\) Give a brief interpretation of the confidence interval. (c) Check Requirements Do you think that \(n p>5\) and \(n q>5\) are satisfied for this problem? Explain why this would be an important consideration.

Step by step solution

01

Determine the Point Estimate for p

To find the point estimate for the proportion \( p \), we use the formula for the sample proportion \( \hat{p} \), defined as \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of traditional hogans and \( n \) is the total number of dwellings sampled. Here, \( x = 1619 \) and \( n = 5222 \). So, \( \hat{p} = \frac{1619}{5222} \approx 0.3102 \).

02

Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \). Substituting the values, we get \( SE = \sqrt{\frac{0.3102 \times 0.6898}{5222}} \approx 0.0064 \).

03

Find the Z-score for a 99% Confidence Interval

For a 99% confidence interval, the Z-score is approximately 2.576. This value corresponds to the critical value associated with the 99% confidence level in a standard normal distribution.

04

Calculate the Margin of Error

The margin of error (ME) is calculated as \( ME = Z \times SE \). Here, \( ME = 2.576 \times 0.0064 \approx 0.0165 \).

05

Determine the Confidence Interval

The confidence interval (CI) is given by \( \hat{p} \pm ME \). Therefore, the 99% confidence interval for \( p \) is \( 0.3102 \pm 0.0165 \), which simplifies to \( (0.2937, 0.3267) \).

06

Check Requirements for Normal Approximation

We verify if the assumptions of the normal approximation are met by checking \( np > 5 \) and \( nq > 5 \). We have \( np = 5222 \times 0.3102 = 1619 \) and \( nq = 5222 \times (1 - 0.3102) = 3603 \). Both values are much greater than 5, hence confirming the requirements are satisfied.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Understanding Point Estimate

A point estimate provides a single-valued estimate of an unknown population parameter. In simple terms, a point estimate uses sample data to give the best guess of the actual value of the population parameter. For the problem of determining the proportion of traditional Navajo hogans, we use the sample proportion, denoted by \( \hat{p} \).
To calculate \( \hat{p} \), we divide the number of hogans in the sample by the total number of dwellings sampled. In our case, the point estimate is given by the formula \( \hat{p} = \frac{1619}{5222} \approx 0.3102 \).
This means approximately 31.02% of the sampled dwellings are traditional hogans.

Insight into Confidence Intervals

A confidence interval offers a range of values within which the true population parameter is expected to fall. Unlike a point estimate, which gives a single value, a confidence interval provides a span, enhancing our understanding by indicating uncertainty.
The 99% confidence interval is calculated using the sample proportion, the Z-score for the desired confidence level, and the standard error. Here, the 99% confidence interval is \( 0.3102 \pm 0.0165 \)
which results in the interval \( (0.2937, 0.3267) \).
This means we can be 99% confident that between 29.37% and 32.67% of all dwellings are traditional hogans.

Exploring Normal Approximation

Normal approximation allows us to use the normal distribution to approximate the distribution of sample proportions, which simplifies calculations.
This approximation is valid under certain conditions, known as the rule of thumb:

• The sample size should be large enough such that both \( np > 5 \) and \( nq > 5 \).
• For our example, with \( np = 1619 \) and \( nq = 3603 \), both conditions are satisfied.

This confirms the use of normal approximation is appropriate, ensuring the reliability of our confidence interval calculations.

Comprehending Sample Proportion

Sample proportion refers to the fraction of the sample that exhibits a particular characteristic. It serves as the point estimate of the population proportion.
In this situation, the sample proportion is calculated as \( \hat{p} = \frac{1619}{5222} \approx 0.3102 \).
This estimation helps to predict the proportion of traditional hogans across the entire Navajo reservation.
The use of sample proportion is crucial because it leverages smaller subset data to make inferences about a larger population.