Expected values

Definition

Expected values represent the average outcome or the mean of a random variable in probability and statistics. They provide a way to quantify the long-term average of a random process, allowing for comparisons between different scenarios or models. In the context of a Chi Square Goodness of Fit Test, expected values help assess how well observed data aligns with the expected distribution under the null hypothesis.

5 Must Know Facts For Your Next Test

1. In a Chi Square Goodness of Fit Test, expected values are calculated based on the null hypothesis and represent the frequencies we would expect to see if the null hypothesis is true.
2. The formula for calculating expected values is: $$E_i = (n) \times (p_i)$$, where $$E_i$$ is the expected frequency for category i, n is the total sample size, and $$p_i$$ is the theoretical proportion for category i.
3. If the expected value for any category is less than 5, it can lead to inaccurate results in a Chi Square test, which may require combining categories to meet this criterion.
4. The chi-square statistic is calculated by comparing the sum of squared differences between observed and expected values divided by the expected values for each category.
5. Expected values are crucial in determining whether there is a significant difference between the observed and expected frequencies, helping to inform conclusions about the data.

Review Questions

• How do you calculate expected values in a Chi Square Goodness of Fit Test, and why are they important?
  • Expected values are calculated using the formula $$E_i = (n) \times (p_i)$$, where $$E_i$$ is the expected frequency for each category, n is the total number of observations, and $$p_i$$ is the expected proportion for that category based on the null hypothesis. They are important because they serve as a baseline against which observed frequencies are compared. By assessing how closely observed data aligns with these expected values, we can determine if there are significant differences that warrant further investigation.
• What might be the implications if some expected values in a Chi Square Goodness of Fit Test are less than 5?
  • If some expected values are less than 5, it can lead to inaccurate results from the Chi Square test because small expected frequencies may not provide reliable estimates of variability. To address this issue, researchers often combine categories or use alternative statistical methods to ensure all expected frequencies meet this minimum threshold. This ensures that the test results will be valid and interpretable.
• Evaluate how expected values impact decision-making in hypothesis testing using a Chi Square Goodness of Fit Test.
  • Expected values significantly influence decision-making in hypothesis testing by providing a standard against which observed data can be measured. When comparing these two sets of values, researchers can assess whether deviations from expectations are statistically significant or simply due to random chance. A significant difference suggests that the observed data does not fit well with what was anticipated under the null hypothesis, prompting further investigation or rejection of that hypothesis. Thus, expected values play a critical role in determining whether to accept or reject initial assumptions about data distributions.

Related terms

Null Hypothesis:

A statement suggesting that there is no effect or no difference, serving as a starting point for statistical testing.

Chi-Square Statistic:

A measure that describes how much the observed frequencies differ from the expected frequencies, used in chi-square tests.

Degrees of Freedom:

The number of independent values or quantities that can be assigned to a statistical distribution; used to determine the appropriate chi-square distribution.