key term - $O_i$

Definition

$O_i$ represents the observed frequency or count of observations in the $i$-th category or bin of a dataset. It is a crucial term in the context of the Goodness-of-Fit Test, which is used to determine whether a dataset follows a hypothesized probability distribution.

5 Must Know Facts For Your Next Test

1. $O_i$ represents the actual count or frequency of observations in the $i$-th category or bin of the dataset.
2. The Goodness-of-Fit Test compares the observed frequencies ($O_i$) to the expected frequencies ($E_i$) to determine if the dataset follows a hypothesized probability distribution.
3. The test statistic ($ extbackslash chi^2$) is calculated using the observed ($O_i$) and expected ($E_i$) frequencies, and is used to determine the p-value for the test.
4. The number of degrees of freedom for the Goodness-of-Fit Test is equal to the number of categories or bins minus the number of parameters estimated from the data.
5. If the p-value from the Goodness-of-Fit Test is less than the chosen significance level, the null hypothesis (that the dataset follows the hypothesized distribution) is rejected.

Review Questions

• Explain the role of $O_i$ in the Goodness-of-Fit Test and how it is used to calculate the test statistic.
  • In the Goodness-of-Fit Test, $O_i$ represents the observed frequency or count of observations in the $i$-th category or bin of the dataset. The test statistic, $ extbackslash chi^2$, is calculated as the sum of the squared differences between the observed frequencies ($O_i$) and the expected frequencies ($E_i$), divided by the expected frequencies ($E_i$). This comparison of the observed and expected frequencies is used to determine whether the dataset follows the hypothesized probability distribution.
• Describe the relationship between $O_i$, $E_i$, and the number of degrees of freedom in the Goodness-of-Fit Test.
  • The Goodness-of-Fit Test compares the observed frequencies ($O_i$) to the expected frequencies ($E_i$) to determine if the dataset follows a hypothesized probability distribution. The number of degrees of freedom for the test is equal to the number of categories or bins minus the number of parameters estimated from the data. This relationship between the observed and expected frequencies, as well as the degrees of freedom, is crucial for calculating the test statistic and determining the p-value to assess the fit of the hypothesized distribution to the dataset.
• Analyze the implications of the p-value from the Goodness-of-Fit Test in relation to the observed frequencies ($O_i$) and the hypothesized probability distribution.
  • The p-value from the Goodness-of-Fit Test is used to determine whether the null hypothesis (that the dataset follows the hypothesized probability distribution) should be rejected or not. If the p-value is less than the chosen significance level, the null hypothesis is rejected, indicating that the observed frequencies ($O_i$) do not match the expected frequencies ($E_i$) based on the hypothesized distribution. This suggests that the dataset does not follow the hypothesized probability distribution, and further investigation may be needed to identify an alternative distribution that better fits the observed data.

Related terms

Goodness-of-Fit Test:

A statistical test used to determine whether a dataset follows a hypothesized probability distribution by comparing the observed frequencies to the expected frequencies.

Expected Frequency ($E_i$): The expected frequency or count of observations in the $i$-th category or bin, calculated based on the hypothesized probability distribution.

Test Statistic ($ extbackslash chi^2$): The test statistic used in the Goodness-of-Fit Test, which is calculated as the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies.