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
a. The wealth of the parents of students attending your school:
In this case, we would not expect the distribution to be normally distributed because wealth typically follows a skewed distribution. There are often a few individuals with extremely high wealth levels, while the majority of individuals have lower levels of wealth. This results in a distribution that is skewed to the right, with a long tail representing the few individuals with high wealth.
To see if the distribution is normally distributed, you would need to collect data on the wealth of parents attending your school and create a histogram or a box plot. These visualizations can help you observe the shape of the distribution. Additionally, you can use statistical tests such as the Shapiro-Wilk test, which assesses the normality of a distribution. By analyzing the p-value from this test, you can determine whether the data significantly deviates from a normal distribution.
b. The values that a group of fourth-grade students would give for the length of a segment they measured with a ruler:
Similarly, we would not expect this distribution to be normally distributed. Fourth-grade students are at a stage in their educational development where they are learning measurement skills and may not have precise control over measurement tools like rulers. As a result, their measurements are likely to be affected by random errors, leading to variability in their responses.
To determine the distribution of these measurements, you would collect data on the length of segments measured by fourth-grade students and create a histogram or a box plot. Again, you can use statistical tests such as the Shapiro-Wilk test to assess the normality of the distribution.
c. The SAT or ACT examination scores in mathematics for students who were in your high school graduation class:
In this case, we would expect the distribution to be approximately normally distributed. Standardized tests like the SAT or ACT are designed to follow a normal distribution, with most students centered around the average score and fewer students achieving extremely high or low scores. This assumption is based on the process of score scaling and equating, which involves adjusting test scores to conform to a normal distribution.
To assess the normality of SAT or ACT scores, you can collect data on the mathematics scores of students in your high school graduation class and create a histogram or a box plot. You can also use statistical tests such as the Shapiro-Wilk test to evaluate the normality of the distribution.
d. The weights of all incoming freshman students at your school:
In this case, we would expect the distribution to be approximately normally distributed. Human body weights tend to follow a bell-shaped curve when considering a large population. This assumption is based on the factors that influence body weight, such as genetics, lifestyle, and nutrition, which collectively tend to lead to a normal distribution.
To determine the distribution of weights for incoming freshman students, you would collect data on their weights and create a histogram or a box plot. As mentioned earlier, statistical tests like the Shapiro-Wilk test can be used to verify the normality of the distribution.