$E_i$ 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.
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$E_i$ is calculated by multiplying the total sample size by the proportion expected under the null hypothesis for each category.
In a goodness-of-fit test, the sum of all expected frequencies ($E_i$) should equal the total number of observations.
The accuracy of $E_i$ affects the validity of the chi-square test; low expected frequencies can lead to inaccurate results.
It’s essential that each $E_i$ is 5 or more for valid conclusions in the chi-square goodness-of-fit test.
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.
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.