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The following data are based on information taken from the book Navajo Architecture: Forms, History, Distributions, by S. C. Jett and V. E. Spencer (University of Arizona Press). A survey of houses and traditional hogans was made in a number of different regions of the modern Navajo Indian Reservation. The following table is the result of a random sample of eight regions on the Navajo Reservation. \(\begin{array}{lcc} \hline \begin{array}{l} \text { Area on } \\ \text { Navajo Reservation } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Inhabited Houses } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Inhabited Hogans } \end{array} \\ \hline \text { Bitter Springs } & 18 & 13 \\ \text { Rainbow Lodge } & 16 & 14 \\ \text { Kayenta } & 68 & 46 \\ \text { Red Mesa } & 9 & 32 \\ \text { Black Mesa } & 11 & 15 \\ \text { Canyon de Chelly } & 28 & 47 \\ \text { Cedar Point } & 50 & 17 \\ \text { Burnt Water } & 50 & 18 \\ \hline \end{array}\) Does this information indicate that the population mean number of inhabited houses is greater than that of hogans on the Navajo Reservation? Use a \(5 \%\) level of significance.

Step by step solution

01

Formulating Hypotheses

We start by setting up our null and alternative hypotheses. The null hypothesis, \(H_0\), states that the mean number of inhabited houses, \(\mu_1\), is equal to the mean number of inhabited hogans, \(\mu_2\). The alternative hypothesis, \(H_a\), asserts that \(\mu_1\) is greater than \(\mu_2\). Mathematically, these are expressed as: \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 > \mu_2\).

02

Calculate Sample Means and Differences

For each region, calculate the difference between the number of inhabited houses and hogans. Then, find the sample mean of these differences: \(d_i = \text{houses} - \text{hogans}\). Calculate the mean of these differences \(\bar{d}\) across all eight regions.

03

Compute Standard Deviation of Differences

Calculate the standard deviation \(s_d\) of the differences \(d_i\) from Step 2. This measures the variability in the differences between houses and hogans across the regions.

04

Conduct t-test for Paired Sample

We use a paired t-test to determine if the mean difference \(\bar{d}\) is significantly greater than zero. This involves the formula: \( t = \frac{\bar{d}}{s_d/\sqrt{n}} \), where \(n\) is the number of regions (which is 8). Compute the t-statistic using the values from Steps 2 and 3.

05

Determine Critical Value and Compare

At a 5% level of significance and based on a one-tailed test with \(n-1=7\) degrees of freedom, find the critical value from the t-distribution table. Compare the computed t-statistic from Step 4 with this critical value.

06

Conclusion

If the computed t-statistic is greater than the critical value, reject the null hypothesis \(H_0\) and accept the alternative hypothesis \(H_a\). This would indicate that the mean number of inhabited houses is indeed greater than that of hogans on the Navajo Reservation.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a powerful statistical method used to make decisions or inferences about population parameters based on sample data. It begins with stating two hypotheses:
- The **null hypothesis** \(H_0\) is a statement of no effect or no difference. In our case, it claims that the average number of inhabited houses on the Navajo Reservation is the same as the average number of inhabited hogans.
- The **alternative hypothesis** \(H_a\), suggests the opposite of the null. Here, it asserts that the average number of houses is greater than that of hogans.

Once the hypotheses are defined, we collect sample data to test them. Statistical tests, like the paired t-test, help determine whether we have enough evidence to reject the null hypothesis in favor of the alternative. This decision is not made lightly, as it involves comparing calculated test statistics to critical values determined by the chosen level of significance. If the test statistic exceeds the critical value, it provides evidence to reject \(H_0\) and accept \(H_a\).

Sample Mean

The sample mean is a fundamental statistic that estimates the central value of a sample data set. It helps in understanding the average behavior of observations.
In our analysis, we calculated the sample mean of the differences between the number of inhabited houses and hogans.- First, find the difference between houses and hogans for each region \(d_i = ext{houses} - ext{hogans}\).- Then, compute the average of these differences to get the sample mean \(\overline{d}\).
This average difference helps us assess whether houses consistently outnumber hogans across the sampled regions. The sample mean is crucial in the paired t-test, representing the observed effect size we evaluate against the null hypothesis.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are around the mean. In our study, we compute the standard deviation of the differences between the number of inhabited houses and hogans. This gives us an idea of the variability or spread of these differences across different regions.
Steps include:

• Subtract the sample mean difference from each individual difference.


• Square these outcomes to eliminate negative values.


• Compute the average of these squared differences.


• Finally, take the square root of this average to obtain the standard deviation \(s_d\).

Understanding standard deviation is vital since it helps in calculating the test statistic for the paired t-test. A smaller standard deviation indicates that the differences are consistently near the average, providing stronger evidence against the null hypothesis if a significant effect is detected.

Level of Significance

The level of significance is a threshold set by the researcher, which determines how extreme the test statistic must be before we can reject the null hypothesis. It is denoted by \( \alpha \), and commonly set at 0.05, 0.01, or 0.10.

In our scenario, we used a 5% level of significance \(\alpha = 0.05\). This means if the probability of observing the test statistic is less than 5%, given the null hypothesis is true, we reject \(H_0\).
This chosen level is crucial because it balances the risk of two types of errors:

• **Type I error**: Rejecting a true null hypothesis.
• **Type II error**: Failing to reject a false null hypothesis.

By carefully selecting \(\alpha\), researchers can control and minimize the chances of making incorrect decisions based on sample data.