How do you calculate the expected value in chi-square?

When conducting a chi-square test, one of the key steps is to calculate the expected values. The expected value represents the number of observations that would be expected in each category if the null hypothesis were true. In order to calculate the expected value, follow these steps:

1. **Determine the total number of observations in the sample.**
2. **Observe the proportions or probabilities expected under the null hypothesis for each category.**
3. **Multiply the total number of observations by the expected proportions or probabilities to obtain the expected count for each category.**
4. **Repeat this process for all categories.**

For a better understanding, let’s consider an example:

Suppose we have a sample of 200 individuals, and we are interested in whether there is an association between gender and smoking status. We determine the following observed counts:

Example:

| | Non-smoker | Smoker | Total |
|——–|————|——–|——–|
| Male | 70 | 30 | 100 |
| Female | 60 | 40 | 100 |
|——–|————|——–|——–|
| Total | 130 | 70 | 200 |

The next step is to calculate the expected values.

Calculating Expected Values:

We can calculate the expected values by following these steps:

1. **Determine the total number of observations in the sample:** In this case, there are 200 individuals.
2. **Observe the proportions or probabilities expected under the null hypothesis:** Assume that gender and smoking status are independent, and calculate the proportions for each category. For example, the proportion of males is 100/200=0.5, and the proportion of non-smokers is 130/200=0.65.
3. **Multiply the total number of observations by the expected proportions:** For each cell, multiply the row and column totals, then divide by the total number of observations. For example, the expected count for males who are non-smokers is (100 * 130) / 200 = 65.
4. **Repeat this process for all categories:** Calculate the expected values for each cell using the same formula.

Using these steps, we can calculate the expected values for our example:

Expected Values:

| | Non-smoker | Smoker | Total |
|——–|————|——–|——–|
| Male | 65 | 35 | 100 |
| Female | 65 | 35 | 100 |
|——–|————|——–|——–|
| Total | 130 | 70 | 200 |

These expected values represent the number of individuals that would be expected in each category if gender and smoking status were independent. By comparing the observed counts to the expected values, we can determine whether there is a significant deviation from independence using the chi-square test.

Frequently Asked Questions (FAQs)

1. What is the chi-square test?

The chi-square test is a statistical test used to determine whether there is a significant association between two categorical variables.

2. How is the chi-square test used?

The chi-square test is used to analyze data from categorical variables to determine if there is a significant association or difference between them.

3. What does the chi-square test tell us?

The chi-square test tells us whether the observed data significantly deviates from the expected values assuming the null hypothesis is true.

4. What is the null hypothesis in a chi-square test?

The null hypothesis in a chi-square test states that there is no association between the categorical variables being tested.

5. What is the alternative hypothesis in a chi-square test?

The alternative hypothesis in a chi-square test states that there is an association between the categorical variables being tested.

6. How is the chi-square statistic calculated?

The chi-square statistic is calculated by comparing the observed frequencies to the expected frequencies and measuring the discrepancy between them.

7. What is the degree of freedom in a chi-square test?

The degree of freedom in a chi-square test is the number of categories minus 1.

8. How is the p-value determined in a chi-square test?

The p-value is determined by comparing the calculated chi-square statistic to the chi-square distribution with the specified degree of freedom.

9. How do you interpret the p-value in a chi-square test?

If the p-value is less than the specified significance level (e.g., 0.05), it suggests that the observed data is significantly different from the expected values, thus rejecting the null hypothesis.

10. Can the chi-square test be used with continuous data?

No, the chi-square test is specifically designed for categorical data analysis.

11. What are some limitations of the chi-square test?

Some limitations of the chi-square test include the assumption of independence between categories and the requirement for an adequate sample size in each cell.

12. Can the chi-square test be used for more than two categorical variables?

Yes, the chi-square test can be extended to analyze associations between more than two categorical variables using contingency tables.

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