To find the expected values under the null hypothesis, we’ll calculate the expected frequencies for each cell based on the total counts for each injury site and outcome type. Here’s the method:

1. Calculate the Expected Values: The expected value for each cell is calculated as follows: $\text{Expected Value} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}$

   Given dataTo find the expected values under the null hypothesis, we’ll calculate the expected frequencies for each cell based on the total counts for each injury site and outcome type. Here’s the method:

2. Calculate the Expected Values: The expected value for each cell is calculated as follows: $\text{Expected Value} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}$

   Given

   • Grand total (total injuries) = 136
   • Row totals:
     • Body = 61
     • Head/Neck = 44
     • Limb = 31
   • Column totals:
     • Fatal = 110
     • Not fatal = 26

3. Calculate Expected Values for Each Cell:

   • Body - Fatal: $\frac{61 \times 110}{136} = 49.34$
   • Body - Not fatal: $\frac{61 \times 26}{136} = 11.66$
   • Head/Neck - Fatal: $\frac{44 \times 110}{136} = 35.59$
   • Head/Neck - Not fatal: $\frac{44 \times 26}{136} = 8.41$
   • Limb - Fatal: $\frac{31 \times 110}{136} = 25.07$
   • Limb - Not fatal: $\frac{31 \times 26}{136} = 5.93$

These are the expected values listed in the table under the null hypothesis.

3. Compute the $\chi^2$ Statistic: To compute the $\chi^2$ statistic, use the formula: $\chi^2 = \sum \frac{(O - E)^2}{E}$ where $O$ is the observed value and $E$ is the expected value for each cell.

   Would you like me to calculate the $\chi^2$ statistic using the provided observed and expected values?

Additional Questions:

1. Do you want to know the interpretation of the $\chi^2$ result in context?
2. Would you like assistance on determining the degrees of freedom for the test?
3. Are you interested in a step-by-step guide on how to perform this calculation in a spreadsheet or software?
4. Do you need help understanding why expected values are used in hypothesis testing?
5. Would you like an explanation on how the $\chi^2$ distribution works?

Tip:

When performing a $\chi^2$ test, remember that larger values indicate a greater discrepancy between observed and expected frequencies, potentially suggesting that the null hypothesis may not be valid.