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A two-proportion z-test is a statistical method used to determine if there is a significant difference between the proportions of two independent groups. Here, we are interested in comparing the proportions of individuals identifying as two or more races in North Dakota and Nevada using their respective survey results.

To perform this test, both groups need to satisfy certain conditions. Specifically, the expected numbers of successes and failures in each sample must be 5 or greater. This ensures that the distribution can be approximated to a normal one, making the z-test applicable. In this exercise:

• The first group is North Dakota: 9 out of 1000 people reported as being of two or more races.
• The second group is Nevada: 17 out of 500 people reported as being of two or more races.

By conducting the test, we can determine if there's a statistically significant difference between the two states' proportions. If Nevada's proportion is significantly higher, it supports the claim made in the hypothesis test.

The normal distribution is paramount in hypothesis testing due to its unique properties. When sample sizes are large enough, the distribution of sample proportions will approximate a normal distribution.

Why Use Normal Distribution?

Normal distribution allows us to use z-tests, making it easier to calculate probabilities and p-values. It's a symmetric distribution where most data points fall close to the mean.

Application in This Exercise

In this problem, both sample groups met the thresholds for using a normal distribution. For both North Dakota and Nevada, the number of expected successes and failures exceed 5, making the normal approximation valid. This enables us to utilize the two-proportion z-test effectively.

Sample proportions are essential when comparing the characteristics of two groups through hypothesis testing. They give a snapshot of what is observed in the sample compared to what may exist in the broader population.
Sample proportions are calculated by dividing the number of observed successes by the total sample size. In this scenario:

• The North Dakota sample proportion: \( \hat{p}_1 = \frac{9}{1000} = 0.009 \).
• The Nevada sample proportion: \( \hat{p}_2 = \frac{17}{500} = 0.034 \).

These figures help set up the hypothesis test by quantifying the observed differences. The larger difference in sample proportions suggests a potential difference in the population proportions which the hypothesis test seeks to affirm or refute.

Normal approximation simplifies the process of hypothesis testing by enabling the application of normal distribution properties. This approximation is particularly useful for large sample sizes where the Central Limit Theorem applies.
For a valid normal approximation of sample proportions, both sample's expected number of successes and failures should be 5 or greater. In our case:

• North Dakota: 9 successes (\( n_1 \hat{p}_1 = 9 \)) and 991 failures (\( n_1 \hat{q}_1 = 991 \)).
• Nevada: 17 successes (\( n_2 \hat{p}_2 = 17 \)) and 483 failures (\( n_2 \hat{q}_2 = 483 \)).

Both samples meet the criteria; thus, the normal approximation can be confidently used. This simplifies the test computation and enables the direct application of z-scores to assess the hypothesis.