Expected Frequency

Definition

Expected frequency refers to the theoretical number of occurrences of an event in a statistical experiment based on a specific hypothesis, especially in the context of goodness-of-fit tests. It is calculated under the assumption that the null hypothesis is true and provides a baseline for comparing observed frequencies. Understanding expected frequency is crucial for determining how well the observed data fits a particular distribution or model.

5 Must Know Facts For Your Next Test

1. Expected frequencies are computed based on the assumption that the null hypothesis is true, indicating no significant difference between observed and expected data.
2. In goodness-of-fit tests, the sum of the expected frequencies should equal the total number of observations to ensure validity.
3. If any expected frequency is less than 5, it may violate test assumptions, leading to unreliable results.
4. The formula for calculating expected frequency in a goodness-of-fit test is: $$E_i = n \cdot p_i$$ where $$E_i$$ is the expected frequency, $$n$$ is the total number of observations, and $$p_i$$ is the probability of each category.
5. Expected frequency plays a vital role in determining the test statistic for chi-square tests, which helps in deciding whether to reject or fail to reject the null hypothesis.

Review Questions

• How is expected frequency calculated and why is it important in hypothesis testing?
  • Expected frequency is calculated using the formula $$E_i = n \cdot p_i$$, where $$E_i$$ represents the expected frequency for a specific category, $$n$$ is the total number of observations, and $$p_i$$ is the probability of that category under the null hypothesis. It is important in hypothesis testing because it provides a benchmark against which observed frequencies can be compared. If there is a significant discrepancy between observed and expected frequencies, it may suggest that the null hypothesis should be rejected.
• Discuss how expected frequency influences the validity of a chi-square test.
  • Expected frequency influences the validity of a chi-square test significantly. For accurate results, it's essential that each expected frequency is at least 5; otherwise, it can lead to unreliable conclusions. When expected frequencies are too low, it can compromise the chi-square distribution's approximation to normality, making it difficult to interpret the results. Thus, ensuring adequate expected frequencies helps uphold the assumptions underlying this statistical test.
• Evaluate the role of expected frequency in assessing model fit and its implications for real-world data analysis.
  • Expected frequency plays a critical role in assessing model fit by providing a theoretical framework against which actual data can be measured. When analyzing real-world data, if observed frequencies deviate significantly from expected frequencies, it suggests that the model may not adequately represent the underlying process generating the data. This evaluation helps researchers refine their models and leads to better predictions and understandings of complex systems. Consequently, addressing discrepancies can improve decision-making based on statistical analyses in various fields such as economics, healthcare, and social sciences.

Related terms

Null Hypothesis: A statement that there is no effect or no difference, serving as the starting point for statistical testing.

Chi-Square Test: A statistical test used to determine if there is a significant difference between the expected and observed frequencies in categorical data.

Degrees of Freedom: The number of independent values or quantities that can vary in an analysis without breaking any constraints; often used in calculating statistical tests.