$E_i$

Definition

$E_i$ represents the expected frequency of a category in a goodness-of-fit test. It is calculated under the null hypothesis, which states that the observed distribution of data fits a specified theoretical distribution. The expected frequency is crucial because it provides a benchmark against which the observed frequencies can be compared to determine if there is a statistically significant difference between them.

5 Must Know Facts For Your Next Test

1. $E_i$ is calculated by multiplying the total sample size by the proportion expected under the null hypothesis for each category.
2. In a goodness-of-fit test, the sum of all expected frequencies ($E_i$) should equal the total number of observations.
3. The accuracy of $E_i$ affects the validity of the chi-square test; low expected frequencies can lead to inaccurate results.
4. It’s essential that each $E_i$ is 5 or more for valid conclusions in the chi-square goodness-of-fit test.
5. Understanding $E_i$ helps in interpreting the chi-square statistic, as it directly influences whether we reject or fail to reject the null hypothesis.

Review Questions

• How is $E_i$ calculated and why is it important in the context of a goodness-of-fit test?
  • $E_i$ is calculated by taking the total number of observations and multiplying it by the expected proportion for each category based on the null hypothesis. It serves as a reference point to compare with observed frequencies ($O_i$). The importance of $E_i$ lies in its role in determining if there are significant differences between what was observed and what was expected, helping to assess whether the null hypothesis can be rejected.
• Discuss how $E_i$ influences the chi-square statistic and its interpretation during hypothesis testing.
  • $E_i$ directly impacts the chi-square statistic since this statistic is computed using both observed ($O_i$) and expected frequencies ($E_i$) through the formula $$ ext{Chi-Square} = rac{(O_i - E_i)^2}{E_i}$$. A larger difference between $O_i$ and $E_i$, when squared and divided by $E_i$, leads to a higher chi-square value. This value helps determine whether to reject or fail to reject the null hypothesis based on critical values from the chi-square distribution.
• Evaluate the implications of having low expected frequencies ($E_i < 5$) on the results of a goodness-of-fit test.
  • Having low expected frequencies can undermine the reliability of a goodness-of-fit test because it violates one of the assumptions necessary for valid conclusions. If many $E_i$ values are below 5, it suggests that the sample size may be too small or that certain categories have insufficient data. This can lead to inaccurate chi-square statistics and potential misinterpretations of statistical significance, making it difficult to confidently reject or accept the null hypothesis.

Related terms

$O_i$:

The observed frequency of a category in a goodness-of-fit test, representing the actual count from the data.

Chi-Square Statistic:

A statistic used in the goodness-of-fit test that measures how much the observed frequencies differ from the expected frequencies, calculated as $$rac{(O_i - E_i)^2}{E_i}$$.

Null Hypothesis: A statement that assumes no significant difference between observed and expected frequencies, serving as the basis for statistical testing.