Exercise: CLT applicability
Consider the class average in an exam in a few different settings. In all cases, assume that we have a large class consisting of equally well prepared students. Think about the assumptions behind the central limit theorem, and choose the most appropriate response under the given description of the different settings.
1. Consider the class average in an exam of fixed difficulty.
(a) The class average is approximately normal
(b) The class average is not approximately normal because the student scores are strongly dependent
(c) The class average is not approximately normal because the student scores are not identically distributed
2. Consider the class average in an exam that is equally likely to be very easy or very hard.
(a) The class average is approximately normal
(b) The class average is not approximately normal because the student scores are strongly dependent
(c) The class average is not approximately normal because the student scores are not identically distributed
3. Consider the class average if the class if split into two equal size sections. One section gets an easy exam and the other section gets a hard exam.
(a) The class average is approximately normal
(b) The class average is not approximately normal because the student scores are strongly dependent
(c) The class average is not approximately normal because the student scores are not identically distributed
4. Consider the class average if every student is (randomly and independently) given either an easy or a hard exam.
(a) The class average is approximately normal
(b) The class average is not approximately normal because the student scores are strongly dependent
(c) The class average is not approximately normal because the student scores are not identically distributed
1. (a)
2. (b)
3. (a)
4. (a)
1. Since students are equally well-prepared and the difficulty level is fixed, the only randomness in a student's score comes from luck or accidental mistakes of that student. It is then plausible to assume that each student's score will be an independent random variable drawn from the same distribution, and the CLT applies.
2. Here, the score of each student depends strongly on the difficulty level of the exam, which is random but common for all students. This creates a strong dependence between the student scores, and the CLT does not apply.
3. This is more subtle. The scores of the different students are not identically distributed. However, let 𝑌𝑖 be the score of the 𝑖 th student from the first section and let 𝑍𝑖 be the score of the 𝑖 th student in the second section. The class average is the average of the random variables (𝑌𝑖+𝑍𝑖)/2 . Under our assumptions, these latter random variables can be modeled as i.i.d., and the CLT applies.
4. Unlike part (2), here the student scores are i.i.d., and the CLT applies.
1. (a) The class average is approximately normal. The assumptions of the central limit theorem hold because the student scores are independent and identically distributed.
2. (a) The class average is approximately normal. The central limit theorem still applies even if the difficulty of the exam can vary, as long as the student scores are independent and identically distributed.
3. (b) The class average is not approximately normal because the student scores are strongly dependent. The scores of students in each section are not independent since they are taking different exams.
4. (a) The class average is approximately normal. The central limit theorem applies because the student scores are independent and identically distributed, even though some students are taking an easy exam and others are taking a hard exam.
1. (a) The class average is approximately normal: In this setting, the class average in the exam of fixed difficulty is applicable to the central limit theorem. The central limit theorem states that under certain conditions, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the original population distribution. As long as the student scores are independently and identically distributed, the class average will be approximately normal.
2. (a) The class average is approximately normal: In this setting, although the exam difficulty can vary, the condition of equally well-prepared students suggests that the student scores are still independently and identically distributed. Therefore, the class average is applicable to the central limit theorem and will be approximately normal.
3. (b) The class average is not approximately normal because the student scores are strongly dependent: In this setting, the class is split into two sections, and each section receives a different difficulty level. As a result, the scores of students within each section will be strongly dependent on the exam difficulty assigned to their section. The central limit theorem assumes independent observations, so in this case, the class average is not approximately normal.
4. (a) The class average is approximately normal: In this setting, every student is randomly and independently assigned either an easy or a hard exam. As long as the students' scores are independently and identically distributed, the class average will be applicable to the central limit theorem and will be approximately normal. The distribution of the class average is not affected by the random assignment of exam difficulties to individual students.
The central limit theorem (CLT) is a statistical concept that states that the distribution of sample means from a population with any shape approaches a normal distribution as the sample size increases. When considering the class average in an exam, we can assess the applicability of the CLT based on certain assumptions.
Let's analyze each scenario:
1. Consider the class average in an exam of fixed difficulty.
In this case, the assumption of equally well-prepared students suggests that the student scores are independent and identically distributed. Therefore, the class average can be expected to be approximately normal. The answer is (a) The class average is approximately normal.
2. Consider the class average in an exam that is equally likely to be very easy or very hard.
Here, the assumption of equally well-prepared students does not hold, as the difficulty of the exam can vary significantly. The student scores would not be identically distributed, and this violates one of the assumptions behind the CLT. Therefore, the class average is not expected to be approximately normal. The answer is (c) The class average is not approximately normal because the student scores are not identically distributed.
3. Consider the class average if the class is split into two equal size sections. One section gets an easy exam and the other section gets a hard exam.
In this situation, the independence assumption is violated. The student scores in each section would be dependent on the specific exam they received. Thus, the class average is not expected to be approximately normal. The answer is (b) The class average is not approximately normal because the student scores are strongly dependent.
4. Consider the class average if every student is (randomly and independently) given either an easy or a hard exam.
In this scenario, the assumption of randomly and independently assigned exams satisfies the independence requirement. Additionally, each student has an equal chance of receiving an easy or a hard exam, ensuring identical distributions. Therefore, the class average can be expected to be approximately normal. The answer is (a) The class average is approximately normal.
By understanding the assumptions behind the CLT and applying them to different situations, we can determine whether the class average would be approximately normal or not.