Question

If the mean and standard deviation of the number of correctly answered questions in a test given to 4096 students are 2.5 and \(\sqrt{1.875}\) respectively, what is the estimate of the number of candidates answering 5 questions correctly?

• A. 232
• B. 234
• C. 237
• D. 239

Hint:

To estimate the number of students answering a certain number of questions correctly, use the Z-score formula and refer to the Z-table to find the corresponding cumulative probability.

Answer 1

The Correct Option is D

We are given: - Mean (\(\mu\)) = 2.5 - Standard deviation (\(\sigma\)) = \(\sqrt{1.875}\) - Total number of students = 4096 The number of students who answered 5 questions correctly can be estimated using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] where \(X = 5\) (the number of correct answers), \(\mu = 2.5\) (mean), and \(\sigma = \sqrt{1.875}\). First, calculate the standard deviation: \[ \sigma = \sqrt{1.875} \approx 1.3693 \] Next, calculate the Z-score: \[ Z = \frac{5 - 2.5}{1.3693} \approx 1.85 \] Now, use the Z-score to estimate the percentage of students answering 5 questions correctly. From standard Z-tables, the cumulative probability for \(Z = 1.85\) is approximately 0.9678. Thus, the estimated percentage of students answering 5 questions correctly is 96.78%. Now, multiply this percentage by the total number of students: \[ \text{Number of students} = 0.9678 \times 4096 \approx 239 \]