To test the effectiveness of inoculation against cholera, we can use a Chi-Square test. Here are the steps to conduct the test:
Step 1: State the hypotheses:
- Null Hypothesis (H0): Inoculation does not prevent an attack from cholera (inoculation and attack are independent).
- Alternative Hypothesis (H1): Inoculation prevents an attack from cholera (inoculation and attack are dependent).
Step 2: Set the significance level (α):
The significance level (α) is given as 0.05 (5%).
Step 3: Calculate the expected frequencies:
To perform the Chi-Square test, we need to calculate the expected frequencies for each cell. The expected frequency for a cell is given by the formula:
Expected Frequency = (Row Total * Column Total) / Grand Total
Using this formula, we can calculate the expected frequencies for each cell:
Expected Frequency for Inoculated and Attacked = (190 * 170) / 790 ≈ 40.9
Expected Frequency for Inoculated and Not Attacked = (190 * 620) / 790 ≈ 149.1
Expected Frequency for Not Inoculated and Attacked = (600 * 170) / 790 ≈ 129.1
Expected Frequency for Not Inoculated and Not Attacked = (600 * 620) / 790 ≈ 470.9
Step 4: Calculate the Chi-Square test statistic:
The Chi-Square test statistic is calculated using the formula:
χ² = ∑ [(Observed Frequency - Expected Frequency)² / Expected Frequency]
Using the observed and expected frequencies, we can calculate the Chi-Square test statistic:
χ² = [(30 - 40.9)² / 40.9] + [(160 - 149.1)² / 149.1] + [(140 - 129.1)² / 129.1] + [(460 - 470.9)² / 470.9]
Step 5: Determine the critical value:
The critical value for the Chi-Square test depends on the degrees of freedom and the significance level. In this case, we have a 2x2 contingency table, so the degrees of freedom is (2-1) * (2-1) = 1.
Looking up the critical value in the Chi-Square distribution table (or using a statistical software), for 1 degree of freedom and a significance level of 0.05, we find that the critical value is approximately 3.841.
Step 6: Make a decision:
If the calculated Chi-Square test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, calculate the Chi-Square test statistic:
χ² = [(30 - 40.9)² / 40.9] + [(160 - 149.1)² / 149.1] + [(140 - 129.1)² / 129.1] + [(460 - 470.9)² / 470.9]
≈ 2.7
Since the calculated Chi-Square test statistic (2.7) is less than the critical value (3.841), we fail to reject the null hypothesis.
Step 7: State the conclusion:
Based on the data and the Chi-Square test, at a significance level of 0.05, there is insufficient evidence to reject the hypothesis that inoculation does not prevent an attack from cholera.
