The question provides us with information about the test performance of students divided into two groups, P and Q. Both groups have the same number of students, and their marks are evaluated out of 200. We are given the mean and standard deviation for each group and are told that the marks are distributed on a normal distribution. Our task is to identify which statement has the highest probability of being true.

Understanding Key Statistical Concepts

Before analyzing the options, let's briefly understand the key statistical terms mentioned, as they are crucial for understanding student performance in an exam:

• Mean (\(\mu\)): This is the average score of a group of students. It indicates the central tendency of the data distribution. A higher mean generally suggests better overall performance for the group.
• Standard Deviation (\(\sigma\)): This measures the dispersion or spread of the student scores around the mean. A small standard deviation indicates that individual scores are clustered closely around the mean, implying a narrower range of scores for most students. A large standard deviation indicates that student scores are spread out over a wider range.
• Normal Distribution: This is a common type of continuous probability distribution where the data points are symmetrically distributed around the mean, forming a bell-shaped curve. In a perfect normal distribution, the mean, median, and mode are all equal.

Exam Performance Data for Student Groups

Let's list the provided statistics for each student group:

Analyzing Each Probability Statement for Exam Scores

Now, let's evaluate each given option based on the provided statistics and the properties of a normal distribution to determine which statement regarding student performance has the highest probability of being true.

Probability of Statement 1: No student in group Q scored less marks than any student in group P.

• This statement suggests that the minimum score achieved by any student in group Q is higher than or equal to the maximum score achieved by any student in group P.
• Group P has a mean of 105 and a standard deviation of 25. Group Q has a mean of 85 and a standard deviation of 5.
• Since Group P's mean (105) is significantly higher than Group Q's mean (85), it implies that Group P generally performs better. However, because both groups follow a normal distribution (meaning there are scores in the lower and upper tails of each distribution), it is highly improbable that every single student in Group Q scored higher than every single student in Group P. In fact, due to P's higher mean, it's more likely the opposite, but still, perfect separation is very unlikely.
• Therefore, this statement has a very low probability of being true.

Probability of Statement 2: No student in group P scored less marks than any student in group Q.

• This statement implies that the lowest score achieved by any student in group P is higher than or equal to the highest score achieved by any student in group Q.
• Let's consider the approximate range for most scores in a normal distribution (e.g., within \(\pm3\sigma\) of the mean, which covers about 99.7% of the data):
  • For Group P: Approximate range is \(105 \pm 3 \times 25 = 105 \pm 75\), so marks could range from 30 to 180.
  • For Group Q: Approximate range is \(85 \pm 3 \times 5 = 85 \pm 15\), so marks could range from 70 to 100.
• Comparing these ranges, it's clear that scores in Group P can go much lower (e.g., 30) than the highest scores in Group Q (e.g., 100). Therefore, it is quite possible and probable for some students in group P to score less marks (e.g., a student in P scoring 60) than some students in group Q (e.g., a student in Q scoring 95).
• Hence, the statement that no student in group P scored less marks than any student in group Q is highly improbable.

Probability of Statement 3: Most students of group Q scored marks in a narrower range than students in group P.

• This statement directly concerns the spread of the marks, which is measured by the standard deviation.
• For Group Q, the standard deviation (\(\sigma_Q\)) is 5.
• For Group P, the standard deviation (\(\sigma_P\)) is 25.
• Since \(\sigma_Q = 5\) is significantly smaller than \(\sigma_P = 25\), it means that the scores of students in group Q are much more tightly clustered around their mean (85) compared to the scores of students in group P, which are more spread out around their mean (105).
• In a normal distribution, a smaller standard deviation precisely indicates that a higher proportion of data points (i.e., most students) will fall within a smaller, narrower range around the mean. For example, about 68% of scores fall within \(\pm1\sigma\) of the mean:
  • For Group Q: \(85 \pm 1 \times 5\) = 80 to 90 (a range of 10 marks).
  • For Group P: \(105 \pm 1 \times 25\) = 80 to 130 (a range of 50 marks).
• This comparison clearly demonstrates that "most students" in Group Q scored within a much narrower range of marks than those in Group P. Therefore, this statement has the highest probability of being true, directly reflecting the meaning of standard deviation.

Probability of Statement 4: The median of the marks of group P is 100.

• For a perfectly normal distribution, the mean, median, and mode are all equal.
• The question states that the marks were distributed on a normal distribution.
• The given mean of group P (\(\mu_P\)) is 105.
• Therefore, for group P, its median should also be 105, as the mean, median, and mode coincide in a normal distribution.
• Since the statement claims the median is 100, this statement is false based on the properties of normal distribution and the provided mean.

Conclusion on Student Performance Data Analysis

Based on the detailed analysis of each statement concerning student performance, the statement that has the highest probability of being true is the one that correctly interprets the meaning of standard deviation in the context of a normal distribution. Group Q's significantly smaller standard deviation (5) compared to Group P's (25) directly implies that the marks of most students in Group Q are concentrated within a much narrower range around their mean.