Answer :
Certainly! Let's go through the steps for testing the hypotheses using the [tex]$\chi^2$[/tex] goodness of fit test.
### Step 1: Define the hypotheses
- Null hypothesis: [tex]\( H_0 \)[/tex]: [tex]\( p_A = 0.40, p_B = 0.40, p_C = 0.20 \)[/tex]
- Alternative hypothesis: [tex]\( H_a \)[/tex]: The probabilities are not [tex]\( p_A = 0.40, p_B = 0.40, \text{ and } p_C = 0.20 \)[/tex]
### Step 2: Gather the observed counts and expected counts
- Observed counts:
- [tex]\( A = 60 \)[/tex]
- [tex]\( B = 120 \)[/tex]
- [tex]\( C = 20 \)[/tex]
- Total sample size: 200
### Step 3: Calculate the expected counts under [tex]\( H_0 \)[/tex]
- [tex]\( E(A) = 0.40 \times 200 = 80 \)[/tex]
- [tex]\( E(B) = 0.40 \times 200 = 80 \)[/tex]
- [tex]\( E(C) = 0.20 \times 200 = 40 \)[/tex]
### Step 4: Compute the chi-square test statistic
The chi-square test statistic ([tex]\( \chi^2 \)[/tex]) is calculated as:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where [tex]\( O_i \)[/tex] are the observed counts, and [tex]\( E_i \)[/tex] are the expected counts.
Substituting the given values:
[tex]\[ \chi^2 = \frac{(60 - 80)^2}{80} + \frac{(120 - 80)^2}{80} + \frac{(20 - 40)^2}{40} \][/tex]
### Step 5: Calculate the degrees of freedom
The degrees of freedom (dof) for this test is [tex]\( \text{number of categories} - 1 = 3 - 1 = 2 \)[/tex].
### Step 6: Use the p-value approach
- The calculated chi-square test statistic is [tex]\( \chi^2 = 35.00 \)[/tex] (to 2 decimals).
- Using a chi-square distribution table or statistical software, the p-value corresponding to [tex]\( \chi^2 = 35.00 \)[/tex] with 2 degrees of freedom is approximately [tex]\( p = 2.51 \times 10^{-8} \)[/tex].
Given the significance level [tex]\( \alpha = 0.01 \)[/tex]:
- [tex]\( p \)[/tex]-value = [tex]\( 2.51 \times 10^{-8} \)[/tex]
### Conclusion using the p-value approach:
Since the p-value ([tex]\( 2.51 \times 10^{-8} \)[/tex]) is less than [tex]\( \alpha = 0.01 \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### Step 7: Use the critical value approach
- The critical value for a chi-square distribution with 2 degrees of freedom at [tex]\( \alpha = 0.01 \)[/tex] is approximately [tex]\( 9.210 \)[/tex] (to 3 decimals).
### Conclusion using the critical value approach:
- Calculated [tex]\(\chi^2\)[/tex] value is [tex]\( 35.00 \)[/tex] (to 2 decimals).
- Critical value [tex]\( x_{0.01}^2 = 9.210 \)[/tex].
Since [tex]\( \chi^2 = 35.00 \)[/tex] is greater than the critical value [tex]\( 9.210 \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### Summary of Conclusions:
- P-value approach: Reject [tex]\( H_0 \)[/tex] because the p-value is less than [tex]\( \alpha = 0.01 \)[/tex].
- Critical value approach: Reject [tex]\( H_0 \)[/tex] because [tex]\( \chi^2 \)[/tex] is greater than the critical value.
Thus, we conclude that there is significant evidence to suggest that the probabilities are not [tex]\( p_A = 0.40, p_B = 0.40, \text{ and } p_C = 0.20 \)[/tex].
### Step 1: Define the hypotheses
- Null hypothesis: [tex]\( H_0 \)[/tex]: [tex]\( p_A = 0.40, p_B = 0.40, p_C = 0.20 \)[/tex]
- Alternative hypothesis: [tex]\( H_a \)[/tex]: The probabilities are not [tex]\( p_A = 0.40, p_B = 0.40, \text{ and } p_C = 0.20 \)[/tex]
### Step 2: Gather the observed counts and expected counts
- Observed counts:
- [tex]\( A = 60 \)[/tex]
- [tex]\( B = 120 \)[/tex]
- [tex]\( C = 20 \)[/tex]
- Total sample size: 200
### Step 3: Calculate the expected counts under [tex]\( H_0 \)[/tex]
- [tex]\( E(A) = 0.40 \times 200 = 80 \)[/tex]
- [tex]\( E(B) = 0.40 \times 200 = 80 \)[/tex]
- [tex]\( E(C) = 0.20 \times 200 = 40 \)[/tex]
### Step 4: Compute the chi-square test statistic
The chi-square test statistic ([tex]\( \chi^2 \)[/tex]) is calculated as:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where [tex]\( O_i \)[/tex] are the observed counts, and [tex]\( E_i \)[/tex] are the expected counts.
Substituting the given values:
[tex]\[ \chi^2 = \frac{(60 - 80)^2}{80} + \frac{(120 - 80)^2}{80} + \frac{(20 - 40)^2}{40} \][/tex]
### Step 5: Calculate the degrees of freedom
The degrees of freedom (dof) for this test is [tex]\( \text{number of categories} - 1 = 3 - 1 = 2 \)[/tex].
### Step 6: Use the p-value approach
- The calculated chi-square test statistic is [tex]\( \chi^2 = 35.00 \)[/tex] (to 2 decimals).
- Using a chi-square distribution table or statistical software, the p-value corresponding to [tex]\( \chi^2 = 35.00 \)[/tex] with 2 degrees of freedom is approximately [tex]\( p = 2.51 \times 10^{-8} \)[/tex].
Given the significance level [tex]\( \alpha = 0.01 \)[/tex]:
- [tex]\( p \)[/tex]-value = [tex]\( 2.51 \times 10^{-8} \)[/tex]
### Conclusion using the p-value approach:
Since the p-value ([tex]\( 2.51 \times 10^{-8} \)[/tex]) is less than [tex]\( \alpha = 0.01 \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### Step 7: Use the critical value approach
- The critical value for a chi-square distribution with 2 degrees of freedom at [tex]\( \alpha = 0.01 \)[/tex] is approximately [tex]\( 9.210 \)[/tex] (to 3 decimals).
### Conclusion using the critical value approach:
- Calculated [tex]\(\chi^2\)[/tex] value is [tex]\( 35.00 \)[/tex] (to 2 decimals).
- Critical value [tex]\( x_{0.01}^2 = 9.210 \)[/tex].
Since [tex]\( \chi^2 = 35.00 \)[/tex] is greater than the critical value [tex]\( 9.210 \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].
### Summary of Conclusions:
- P-value approach: Reject [tex]\( H_0 \)[/tex] because the p-value is less than [tex]\( \alpha = 0.01 \)[/tex].
- Critical value approach: Reject [tex]\( H_0 \)[/tex] because [tex]\( \chi^2 \)[/tex] is greater than the critical value.
Thus, we conclude that there is significant evidence to suggest that the probabilities are not [tex]\( p_A = 0.40, p_B = 0.40, \text{ and } p_C = 0.20 \)[/tex].
