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Hypothesis testing is a fundamental aspect of statistics that helps us make decisions based on data analysis. In the context of this exercise, we want to determine if a passenger's survival on the Titanic was influenced by the type of ticket they held. To do this, we set up two hypotheses:

• The **null hypothesis** (\(H_0\)) suggests that survival is independent of the ticket class. In other words, the class of ticket a passenger had does not influence their likelihood of survival.
• The **alternative hypothesis** (\(H_1\)) suggests that survival depends on the ticket class.

If evidence strongly contradicts the null hypothesis, we reject it in favor of the alternative hypothesis. This evidence is judged through statistical tests, which calculate probabilities (like the p-value) to guide decision-making. A common threshold used to determine statistical significance is \(\alpha = 0.05\), meaning there's only a 5% chance that the observed data could occur if the null hypothesis is true. Initiating hypothesis testing involves these logical steps and calculations to help us interpret complex data.

In statistics, degrees of freedom (\(df\)) refers to the number of values in a calculation that are free to vary. It's crucial in determining the validity of statistical tests like the Chi-square test. In the Titanic example, we calculate degrees of freedom to determine the appropriate Chi-square distribution against which the test statistic will be compared. To compute this, the formula for degrees of freedom in a contingency table is \((\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)\).Given:

• Number of rows: 2 (representing "Alive" and "Dead" categories)
• Number of columns: 4 (representing Crew, First, Second, and Third ticket classes)

This gives us degrees of freedom as \((2-1)(4-1) = 3\). This degree of freedom is the basis for determining the critical value in the Chi-square distribution, which in turn helps us decide if our observed results significantly differ from the expected ones.

The p-value is a metric used to determine the significance of statistical results in hypothesis testing. It helps us understand the strength of evidence against the null hypothesis. For the Titanic survival data, after calculating the Chi-square test statistic, we find the p-value to understand how likely it is to observe such extreme data if the null hypothesis were true. The smaller the p-value, the stronger the evidence against the null hypothesis.Here's how it is used:

• If the p-value is less than or equal to \(\alpha = 0.05\), the evidence against the null hypothesis is strong enough to reject it. This suggests a relationship between ticket class and survival.
• If the p-value is greater than \(\alpha = 0.05\), it indicates insufficient evidence to reject the null hypothesis, pointing to an assumption that ticket class and survival are independent.

A proper understanding and interpretation of the p-value are key in drawing meaningful conclusions from hypothesis tests.

Expected frequency is the hypothesized distribution of data across different categories if the null hypothesis were true. It aids in determining if observed data significantly deviates from what is "expected" when variables are independent. Here's how to compute it:The formula for expected frequency in a contingency table is \(E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}\).For each cell of the Titanic data table, calculate expected frequencies to compare against observed data:

• For "Alive, Crew" class: \(E = \frac{(710)(885)}{2201} \approx 285.37\)
• Repeat this for all other combinations in the table.

Comparing expected frequencies with observed frequencies using the Chi-square formula helps determine how divergent our data are from what might be expected if survival and ticket class were independent. This process is central to evaluating whether the data supports rejecting or retaining the null hypothesis.