Answer :
Sure, let's go through each part of the problem step-by-step.
### Part A: Percentage of students who failed the exam, given they completed the exam review
To find the percentage of students who failed the exam, given that they completed the exam review, we need to look at the students who completed the review and determine the proportion of those who did not pass.
From the table:
- Percentage of students who completed the exam review: [tex]\(65\%\)[/tex]
- Percentage of students who completed the review and failed: [tex]\(10\%\)[/tex]
We use the conditional probability formula here:
[tex]\[ P(\text{Failed} \mid \text{Reviewed}) = \frac{\text{Percentage of students who completed the review and failed}}{\text{Percentage of students who completed the review}} \times 100 \][/tex]
Substituting the given values:
[tex]\[ P(\text{Failed} \mid \text{Reviewed}) = \frac{10\%}{65\%} \times 100 \approx 15.38\% \][/tex]
After rounding to the nearest whole percentage, we get:
[tex]\[ P(\text{Failed} \mid \text{Reviewed}) \approx 15\% \][/tex]
### Part B: Percentage of students who failed the exam, given they did not complete the exam review
Next, we want to determine the percentage of students who failed the exam among those who did not complete the review.
From the table:
- Percentage of students who did not complete the exam review: [tex]\(36\%\)[/tex]
- Percentage of students who did not complete the review and failed: [tex]\(15\%\)[/tex]
Applying the same conditional probability formula:
[tex]\[ P(\text{Failed} \mid \text{Not Reviewed}) = \frac{\text{Percentage of students who did not complete the review and failed}}{\text{Percentage of students who did not complete the review}} \times 100 \][/tex]
Substituting the given values:
[tex]\[ P(\text{Failed} \mid \text{Not Reviewed}) = \frac{15\%}{36\%} \times 100 \approx 41.67\% \][/tex]
After rounding to the nearest whole percentage, we get:
[tex]\[ P(\text{Failed} \mid \text{Not Reviewed}) \approx 42\% \][/tex]
### Part C: Association between failing the exam and completing the exam review
To determine if there is an association between failing the exam and completing/not completing the exam review, we compare the failure rates of the two groups:
- Failure rate of those who completed the review: [tex]\(15\%\)[/tex]
- Failure rate of those who did not complete the review: [tex]\(42\%\)[/tex]
Since there is a noticeable difference in failure rates (15% vs 42%), we can infer that there is an association between completing the review and failing the exam.
If the failure rates were the same or very similar, we could conclude there is no association. However, because the rates differ significantly, there appears to be an association.
### Summary:
- Part A: The percentage of students who failed the exam, given they completed the exam review, is approximately [tex]\(15\%\)[/tex].
- Part B: The percentage of students who failed the exam, given they did not complete the exam review, is approximately [tex]\(42\%\)[/tex].
- Part C: Yes, there is an association between failing the exam and completing the exam review, as evidenced by the differing failure rates.
### Part A: Percentage of students who failed the exam, given they completed the exam review
To find the percentage of students who failed the exam, given that they completed the exam review, we need to look at the students who completed the review and determine the proportion of those who did not pass.
From the table:
- Percentage of students who completed the exam review: [tex]\(65\%\)[/tex]
- Percentage of students who completed the review and failed: [tex]\(10\%\)[/tex]
We use the conditional probability formula here:
[tex]\[ P(\text{Failed} \mid \text{Reviewed}) = \frac{\text{Percentage of students who completed the review and failed}}{\text{Percentage of students who completed the review}} \times 100 \][/tex]
Substituting the given values:
[tex]\[ P(\text{Failed} \mid \text{Reviewed}) = \frac{10\%}{65\%} \times 100 \approx 15.38\% \][/tex]
After rounding to the nearest whole percentage, we get:
[tex]\[ P(\text{Failed} \mid \text{Reviewed}) \approx 15\% \][/tex]
### Part B: Percentage of students who failed the exam, given they did not complete the exam review
Next, we want to determine the percentage of students who failed the exam among those who did not complete the review.
From the table:
- Percentage of students who did not complete the exam review: [tex]\(36\%\)[/tex]
- Percentage of students who did not complete the review and failed: [tex]\(15\%\)[/tex]
Applying the same conditional probability formula:
[tex]\[ P(\text{Failed} \mid \text{Not Reviewed}) = \frac{\text{Percentage of students who did not complete the review and failed}}{\text{Percentage of students who did not complete the review}} \times 100 \][/tex]
Substituting the given values:
[tex]\[ P(\text{Failed} \mid \text{Not Reviewed}) = \frac{15\%}{36\%} \times 100 \approx 41.67\% \][/tex]
After rounding to the nearest whole percentage, we get:
[tex]\[ P(\text{Failed} \mid \text{Not Reviewed}) \approx 42\% \][/tex]
### Part C: Association between failing the exam and completing the exam review
To determine if there is an association between failing the exam and completing/not completing the exam review, we compare the failure rates of the two groups:
- Failure rate of those who completed the review: [tex]\(15\%\)[/tex]
- Failure rate of those who did not complete the review: [tex]\(42\%\)[/tex]
Since there is a noticeable difference in failure rates (15% vs 42%), we can infer that there is an association between completing the review and failing the exam.
If the failure rates were the same or very similar, we could conclude there is no association. However, because the rates differ significantly, there appears to be an association.
### Summary:
- Part A: The percentage of students who failed the exam, given they completed the exam review, is approximately [tex]\(15\%\)[/tex].
- Part B: The percentage of students who failed the exam, given they did not complete the exam review, is approximately [tex]\(42\%\)[/tex].
- Part C: Yes, there is an association between failing the exam and completing the exam review, as evidenced by the differing failure rates.
