In each case, consider what you know about the distribution

and then explain why you would expect it to be
or not to be normally distributed.

a. The wealth of the parents of students attending your
school
b. The values that a group of fourth-grade students
would give for the length of a segment that they
measured with a ruler
c. The SAT or ACT examination scores in mathematics
for students who were in your high school
graduation class
d. The weights of all incoming freshman students at
your school

*I have no idea how to even start answering these problems*

This site may help you understand normal distribution.

http://www.netmba.com/statistics/distribution/normal/

The first one (letter a.) would depend upon the school. Some schools have many well-to-do families and only a few low-income famiies. Others would have more of a normal distribution of incomes.

I would guess that b. would be normal distribution.

To analyze whether a distribution is expected to be normally distributed, we can consider a few factors:

1. Central Limit Theorem: The Central Limit Theorem states that the sampling distribution of the mean of any independent, random variable will be approximately normally distributed, regardless of the shape of the original population distribution, if the sample size is large enough. Therefore, if the sample size is sufficiently large for each case, we can expect the distribution to be approximately normal.

2. The nature of the variable: The type of variable being considered can also provide insights into its expected distribution. Let's analyze the given scenarios one by one:

a. The wealth of the parents of students attending your school: Wealth is typically not normally distributed because a small proportion of the population holds a significant portion of the wealth. This often results in a skewed distribution, with a long tail on the higher end.

b. The values that a group of fourth-grade students would give for the length of a segment that they measured with a ruler: Since this involves measuring physical objects, it is reasonable to assume that the measurements will be normally distributed, assuming the measurements are accurate and random errors are normally distributed.

c. The SAT or ACT examination scores in mathematics for students who were in your high school graduation class: Exam scores are often assumed to follow a roughly normal distribution, especially if the test is designed to approximate a normal distribution. However, it is worth noting that standardized test scores may have a slight right-skew due to a ceiling effect (maximum achievable score) and the presence of high-achieving outliers.

d. The weights of all incoming freshman students at your school: The distribution of weights is usually bell-shaped and approximately normally distributed. However, there might be slight variations depending on factors such as the demographic composition of the student population.

These are general guidelines, and actual data analysis should be done to determine the specific shape of the distribution using appropriate statistical tests and visualizations.