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Non-parametric statistics are a type of statistical methods that do not assume a specific distribution for the data, such as being normally distributed.
This makes them particularly useful in analyzing ordinal data or when we do not know or cannot assume the underlying distribution of the data.
One popular non-parametric test is the Wilcoxon Rank-Sum Test, which is used to compare two independent samples.
The Wilcoxon Rank-Sum Test operates by ranking all data from both sample groups together, regardless of their origin.
Then, it calculates a statistic based on these ranks to determine if there is a statistically significant difference between the two samples.
This test is particularly useful when dealing with small sample sizes or when data are skewed. It's also robust in handling outliers, which can cause issues in parametric methods. In the context of the crime rate data from New York and New Jersey, the Wilcoxon Rank-Sum Test helps us determine whether the two states have different crime rate distributions without relying on the assumptions of normality.
This makes it a very flexible and widely applicable tool in statistical analysis.

Hypothesis testing is a fundamental concept in statistics that involves making inferences about a population based on a sample.
The process begins by assuming a null hypothesis, which is a statement of no effect or no difference.
This null hypothesis is typically expressed as:

• Null Hypothesis (H0): There is no difference in the distributions between two groups.

An alternative hypothesis is also formulated which expresses the opposite of what is stated in the null hypothesis, such as:

• Alternative Hypothesis (H1): There is a difference in the distributions between two groups.

The objective of hypothesis testing is to use the data collected to evaluate whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.
In our example with New York and New Jersey crime rates, the null hypothesis is that there is no difference in crime rates between the two states. The alternative suggests there is a difference. By calculating a statistical test (in this case, the Wilcoxon Rank-Sum) and comparing it to a critical value from a statistical distribution, we can make a decision about these hypotheses.
If the computed statistic lies in a certain range, we reject the null hypothesis, concluding that there is indeed a difference.

The level of significance, often denoted as alpha (b1), is a threshold that determines the critical value in statistical hypothesis testing.
It quantifies the probability of rejecting the null hypothesis when it is true, known as a Type I error.
Common levels of significance are 0.05, 0.01, and 0.10, with 0.05 being the most frequently used. For the exercise of comparing crime rates between New York and New Jersey, a 5% level of significance is used.
This means that there is a 5% risk of concluding that there is a difference in crime rate distributions when, in reality, there is none.
A low level of significance indicates a higher threshold for evidence, making it less likely to reject the null hypothesis by chance. In hypothesis testing, this level of significance helps determine the critical value that borders the acceptance and rejection regions of the test statistic.
When the test statistic falls within the rejection region, we have enough evidence to reject the null hypothesis.
Therefore, setting and understanding the level of significance is crucial in the hypothesis testing process.